Generalized Motzkin numbers modulo primes

DefinitionThe symmetric trinomial Laurent polynomial#✓fully proved

For integers \(a\) and \(b\), this is the Laurent polynomial \(a x^{-1} + b + a x\) with integer coefficients, used to define the generalized central trinomial coefficients and the generalized Motzkin numbers.

public noncomputable def trinomialLaurent (a b : ℤ) : LaurentPolynomial ℤ :=
  C a * T (-1) + C b + C a * T 1
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Definitions.
Used by 7
Used byOEIS A005717 · OEIS A005773 · Generalized central trinomial coefficients · First five central trinomial coefficients · Generalized Motzkin numbers · First five generalized Motzkin numbers at \(a = b = 1\) · The first five powers of \(x^{-1} + 1 + x\)

DefinitionLaurent polynomials of degree at most one#✓fully proved

For integers \(\alpha, \beta, \gamma\), this is the Laurent polynomial \(\alpha x^{-1} + \beta + \gamma x\) with integer coefficients.

public noncomputable def linearLaurent (α β γ : ℤ) : LaurentPolynomial ℤ :=
  C α * T (-1) + C β + C γ * T 1
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Lemma 1. The degree-one hypotheses of Lemma 1 are imposed on the coefficients in the statements that use this polynomial.
Used by 3
Used byDigit decomposition of shifted sums of constant terms modulo a prime · Digit formula for shifted sums of constant terms, small last digit · Lucas property for constant terms of powers of linear Laurent polynomials

DefinitionGeneralized central trinomial coefficients#✓fully proved

For integers \(a, b\) and a natural number \(n\), the generalized central trinomial coefficient \(T_n^{(a/b)}\) is the constant term of the Laurent polynomial \((a x^{-1} + b + a x)^n\).

public noncomputable def centralTrinomial (a b : ℤ) (n : ℕ) : ℤ :=
  (trinomialLaurent a b ^ n).coeff 0
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Definitions.
Uses 1 · Used by 21
Statement usesThe symmetric trinomial Laurent polynomial
Used byZero density of A005717 modulo a prime · Zero density of A005773 modulo a prime · Mirror symmetry of central trinomial coefficients modulo a prime · First five central trinomial coefficients · Squared central trinomial coefficient at one less than a prime · Recurrence for generalized central trinomial coefficients · Lucas property for constant terms of powers of linear Laurent polynomials · Generalized Motzkin numbers from three central trinomial coefficients · Generalized Motzkin numbers modulo \(p\) with last digit below \(p - 2\) · Generalized Motzkin numbers modulo \(p\) with last digit \(p - 1\) · Generalized Motzkin numbers modulo \(p\) with last digit \(p - 2\) · Density of each nonzero residue among generalized Motzkin numbers modulo \(p\) · Zero density formula for generalized Motzkin numbers modulo a prime · Generalized Motzkin numbers vanish modulo a prime with density one · Zeros of generalized Motzkin numbers with a small last digit · Zeros of generalized Motzkin numbers with last digit \(p - 2\) · Zeros of generalized Motzkin numbers with last digit \(p - 1\) · Zero density of Motzkin numbers modulo a prime · Zero density of Riordan numbers modulo a prime · Zero density of shifted combinations of central trinomial coefficients · Density one for zeros of shifted combinations of central trinomial coefficients

DefinitionGeneralized Motzkin numbers#✓fully proved

For integers \(a, b\) and a natural number \(n\), the generalized Motzkin number \(M^{(a,b)}_n\) is the constant term of the Laurent polynomial \((a x^{-1} + b + a x)^n (1 - x^2)\) with integer coefficients.

public noncomputable def generalizedMotzkin (a b : ℤ) (n : ℕ) : ℤ :=
  (trinomialLaurent a b ^ n * (1 - T 2)).coeff 0
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Definitions.
Uses 1 · Used by 15
Statement usesThe symmetric trinomial Laurent polynomial
Used byGeneralized Motzkin numbers from three central trinomial coefficients · Generalized Motzkin numbers modulo \(p\) with last digit below \(p - 2\) · Generalized Motzkin numbers modulo \(p\) with last digit \(p - 1\) · Generalized Motzkin numbers modulo \(p\) with last digit \(p - 2\) · Reflection symmetry of generalized Motzkin numbers modulo \(p\) · Density of each nonzero residue among generalized Motzkin numbers modulo \(p\) · First five generalized Motzkin numbers at \(a = b = 1\) · Three-term recurrence for generalized Motzkin numbers · Zero density formula for generalized Motzkin numbers modulo a prime · Generalized Motzkin numbers vanish modulo a prime with density one · Zeros of generalized Motzkin numbers with a small last digit · Zeros of generalized Motzkin numbers with last digit \(p - 2\) · Zeros of generalized Motzkin numbers with last digit \(p - 1\) · Zero density of Motzkin numbers modulo a prime · Primes dividing the Motzkin number \(M_{p-2}\)

DefinitionOEIS A005717#✓fully proved

For a natural number \(n\), \(a_n\) is the constant term of the Laurent polynomial \(x (x^{-1} + 1 + x)^n\) with integer coefficients.

public noncomputable def a005717 (n : ℕ) : ℤ :=
  (trinomialLaurent 1 1 ^ n * T 1).coeff 0
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Definitions.
Uses 1 · Used by 1
Statement usesThe symmetric trinomial Laurent polynomial
Used byZero density of A005717 modulo a prime

DefinitionOEIS A005773#✓fully proved

For a natural number \(n\), \(a_n\) is the constant term of the Laurent polynomial \((1 + x)(x^{-1} + 1 + x)^n\) with integer coefficients.

public noncomputable def a005773 (n : ℕ) : ℤ :=
  (trinomialLaurent 1 1 ^ n * (1 + T 1)).coeff 0
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Definitions.
Uses 1 · Used by 1
Statement usesThe symmetric trinomial Laurent polynomial
Used byZero density of A005773 modulo a prime

TheoremThe first five powers of \(x^{-1} + 1 + x\)#✓fully proved

In the ring of Laurent polynomials over \(\mathbb{Z}\), \((x^{-1} + 1 + x)^0 = 1\), \((x^{-1} + 1 + x)^1 = x^{-1} + 1 + x\), \((x^{-1} + 1 + x)^2 = x^{-2} + 2x^{-1} + 3 + 2x + x^2\), \((x^{-1} + 1 + x)^3 = x^{-3} + 3x^{-2} + 6x^{-1} + 7 + 6x + 3x^2 + x^3\) and \((x^{-1} + 1 + x)^4 = x^{-4} + 4x^{-3} + 10x^{-2} + 16x^{-1} + 19 + 16x + 10x^2 + 4x^3 + x^4\).

MetaMathlibExt.trinomialLaurent_one_one_powers · 20 lines 
public theorem trinomialLaurent_one_one_powers :
    trinomialLaurent 1 1 ^ 0 = AddMonoidAlgebra.single 0 1 ∧
    trinomialLaurent 1 1 ^ 1 =
      AddMonoidAlgebra.single (-1) 1 + AddMonoidAlgebra.single 0 1 +
        AddMonoidAlgebra.single 1 1 ∧
    trinomialLaurent 1 1 ^ 2 =
      AddMonoidAlgebra.single (-2) 1 + AddMonoidAlgebra.single (-1) 2 +
        AddMonoidAlgebra.single 0 3 + AddMonoidAlgebra.single 1 2 +
        AddMonoidAlgebra.single 2 1 ∧
    trinomialLaurent 1 1 ^ 3 =
      AddMonoidAlgebra.single (-3) 1 + AddMonoidAlgebra.single (-2) 3 +
        AddMonoidAlgebra.single (-1) 6 + AddMonoidAlgebra.single 0 7 +
        AddMonoidAlgebra.single 1 6 + AddMonoidAlgebra.single 2 3 +
        AddMonoidAlgebra.single 3 1 ∧
    trinomialLaurent 1 1 ^ 4 =
      AddMonoidAlgebra.single (-4) 1 + AddMonoidAlgebra.single (-3) 4 +
        AddMonoidAlgebra.single (-2) 10 + AddMonoidAlgebra.single (-1) 16 +
        AddMonoidAlgebra.single 0 19 + AddMonoidAlgebra.single 1 16 +
        AddMonoidAlgebra.single 2 10 + AddMonoidAlgebra.single 3 4 +
        AddMonoidAlgebra.single 4 1
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Definitions.
Uses 1 · Used by 2
Statement usesThe symmetric trinomial Laurent polynomial
Used byFirst five central trinomial coefficients · First five generalized Motzkin numbers at \(a = b = 1\)

TheoremFirst five central trinomial coefficients#✓fully proved

The central trinomial coefficients \(T_n\), the constant terms of \((x^{-1} + 1 + x)^n\), are \(1, 1, 3, 7, 19\) for \(n = 0, 1, 2, 3, 4\).

public theorem centralTrinomial_one_one_values :
    (List.range 5).map (centralTrinomial 1 1) = [1, 1, 3, 7, 19]
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Definitions.
Uses 3
Statement usesGeneralized central trinomial coefficients
Proof usesThe symmetric trinomial Laurent polynomial · The first five powers of \(x^{-1} + 1 + x\)

TheoremFirst five generalized Motzkin numbers at \(a = b = 1\)#✓fully proved

The generalized Motzkin numbers \(M_n\) for \(a = b = 1\), the constant terms of \((x^{-1} + 1 + x)^n (1 - x^2)\), are \(1, 1, 2, 4, 9\) for \(n = 0, 1, 2, 3, 4\).

public theorem generalizedMotzkin_one_one_values :
    (List.range 5).map (generalizedMotzkin 1 1) = [1, 1, 2, 4, 9]
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Definitions.
Uses 3
Statement usesGeneralized Motzkin numbers
Proof usesThe symmetric trinomial Laurent polynomial · The first five powers of \(x^{-1} + 1 + x\)

TheoremGeneralized Motzkin numbers from three central trinomial coefficients#●ready to prove

For integers \(a\), \(b\) let \(T_r\) and \(M_r\) be the constant terms of \((a x^{-1} + b + a x)^r\) and of \((a x^{-1} + b + a x)^r (1 - x^2)\), the generalized central trinomial coefficients and generalized Motzkin numbers. Then for every natural number \(n\),

\[ 2a^2 M_n = (4a^2 - b^2) T_n + 2b T_{n+1} - T_{n+2}. \]
public theorem_wanted generalizedMotzkin_centralTrinomial_relation
    (a b : ℤ) (n : ℕ) :
    2 * a ^ 2 * generalizedMotzkin a b n =
      (4 * a ^ 2 - b ^ 2) * centralTrinomial a b n +
        2 * b * centralTrinomial a b (n + 1) -
          centralTrinomial a b (n + 2)
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Proposition 1.
Uses 2
Statement usesGeneralized central trinomial coefficients · Generalized Motzkin numbers

TheoremRecurrence for generalized central trinomial coefficients#●ready to prove

Let \(a, b\) be integers and let \(T_n\) be the constant term of \((a x^{-1} + b + a x)^n\). For every natural number \(n\),

\[(n + 2) T_{n+2} = b (2n + 3) T_{n+1} - (b^2 - 4a^2)(n + 1) T_n.\]

This is the source's recurrence with its index shifted by two, so that every index is a natural number.

public theorem_wanted centralTrinomial_recurrence (a b : ℤ) (n : ℕ) :
    (n + 2 : ℤ) * centralTrinomial a b (n + 2) =
      b * (2 * n + 3 : ℤ) * centralTrinomial a b (n + 1) -
        (b ^ 2 - 4 * a ^ 2) * (n + 1 : ℤ) * centralTrinomial a b n
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Proposition 2.
Uses 1
Statement usesGeneralized central trinomial coefficients

TheoremThree-term recurrence for generalized Motzkin numbers#●ready to prove

For integers \(a\), \(b\) let \(M_r\) be the constant term of \((a x^{-1} + b + a x)^r (1 - x^2)\), the generalized Motzkin number. Then for every natural number \(n\),

\[ (n + 4) M_{n+2} = b (2n + 5) M_{n+1} - (b^2 - 4a^2)(n + 1) M_n. \]

The source states the recurrence for \(M_n\) with \(n \ge 2\); it is written here at index \(n + 2\).

public theorem_wanted generalizedMotzkin_recurrence (a b : ℤ) (n : ℕ) :
    (n + 4 : ℤ) * generalizedMotzkin a b (n + 2) =
      b * (2 * n + 5 : ℤ) * generalizedMotzkin a b (n + 1) -
        (b ^ 2 - 4 * a ^ 2) * (n + 1 : ℤ) * generalizedMotzkin a b n
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Corollary 1.
Uses 1
Statement usesGeneralized Motzkin numbers

TheoremMirror symmetry of central trinomial coefficients modulo a prime#●ready to prove

Let \(a, b\) be integers and let \(T_n\) be the constant term of \((a x^{-1} + b + a x)^n\). For every prime \(p > 2\) and every natural number \(k \le (p - 1)/2\),

\[T_{p-1-k} \equiv (b^2 - 4a^2)^{(p-1)/2 - k} \, T_k \pmod p.\]
public theorem_wanted centralTrinomial_mod_prime_symmetry
    (a b : ℤ) (p k : ℕ) (hp : p.Prime) (hp2 : 2 < p)
    (hk : k ≤ (p - 1) / 2) :
    centralTrinomial a b (p - 1 - k) ≡
      (b ^ 2 - 4 * a ^ 2) ^ ((p - 1) / 2 - k) * centralTrinomial a b k
      [ZMOD p]
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Theorem 1.
Uses 1
Statement usesGeneralized central trinomial coefficients

TheoremReflection symmetry of generalized Motzkin numbers modulo \(p\)#●ready to prove

For integers \(a\), \(b\) let \(M_r\) be the constant term of \((a x^{-1} + b + a x)^r (1 - x^2)\), the generalized Motzkin number. Let \(p > 3\) be a prime and \(k\) a natural number with \(k \le (p - 3)/2\). Then

\[ M_{p-3-k} \equiv (b^2 - 4a^2)^{(p-3)/2 - k} \, M_k \pmod p. \]
public theorem_wanted generalizedMotzkin_mod_prime_symmetry
    (a b : ℤ) (p k : ℕ) (hp : p.Prime) (hp3 : 3 < p)
    (hk : k ≤ (p - 3) / 2) :
    generalizedMotzkin a b (p - 3 - k) ≡
      (b ^ 2 - 4 * a ^ 2) ^ ((p - 3) / 2 - k) * generalizedMotzkin a b k
      [ZMOD p]
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Theorem 2.
Uses 1
Statement usesGeneralized Motzkin numbers

TheoremSquared central trinomial coefficient at one less than a prime#●ready to prove

Let \(a, b\) be integers, let \(T_n\) be the constant term of \((a x^{-1} + b + a x)^n\), and let \(p\) be a prime with \(b^2 - 4a^2 \not\equiv 0 \pmod p\). Then \(T_{p-1}^2 \equiv 1 \pmod p\).

This is the source's Corollary 2 in an equivalent squared form, which avoids an inverse residue modulo \(p\).

public theorem_wanted centralTrinomial_prime_minus_one_sq
    (a b : ℤ) (p : ℕ) (hp : p.Prime)
    (hdisc : ¬(b ^ 2 - 4 * a ^ 2 ≡ 0 [ZMOD p])) :
    centralTrinomial a b (p - 1) ^ 2 ≡ 1 [ZMOD p]
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Corollary 2.
Uses 1
Statement usesGeneralized central trinomial coefficients

TheoremPrimes dividing the Motzkin number \(M_{p-2}\)#●ready to prove

A prime \(p\) divides the Motzkin number \(M_{p-2} = M^{(1,1)}_{p-2}\) if and only if \(p \equiv 1 \pmod 3\).

For \(p = 2\) both sides are false, since \(M_0 = 1\).

public theorem_wanted prime_dvd_motzkin_prime_minus_two_iff
    (p : ℕ) (hp : p.Prime) :
    generalizedMotzkin 1 1 (p - 2) ≡ 0 [ZMOD p] ↔ p ≡ 1 [MOD 3]
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Proposition 3.
Uses 1
Statement usesGeneralized Motzkin numbers

TheoremLucas property for constant terms of powers of linear Laurent polynomials#●ready to prove

Let \(p\) be a prime, let \(\alpha\), \(\beta\), \(\gamma\), \(a\), \(b\) be integers, and let \(n\) be a natural number whose base-\(p\) digits are \(d_0, \dots, d_s\) (there are none when \(n = 0\), and both products below are then \(1\)). Write \(\operatorname{ct}(F)\) for the constant term of a Laurent polynomial \(F\) and put \(P = \alpha x^{-1} + \beta + \gamma x\). Then

\[ \operatorname{ct}(P^n) \equiv \prod_{i=0}^{s} \operatorname{ct}(P^{d_i}) \pmod p, \]

and the generalized central trinomial coefficients \(T_r = \operatorname{ct}((a x^{-1} + b + a x)^r)\) satisfy \(T_n \equiv \prod_{i=0}^{s} T_{d_i} \pmod p\).

The source assumes that \(P\) has degree one; here \(\alpha\) and \(\gamma\) are arbitrary, which includes that case.

public theorem_wanted constantTerm_linearLaurent_pow_digits (α β γ a b : ℤ) (p n : ℕ)
    (hp : p.Prime) :
    (linearLaurent α β γ ^ n).coeff 0 ≡
        ((Nat.digits p n).map fun d => (linearLaurent α β γ ^ d).coeff 0).prod [ZMOD p] ∧
      centralTrinomial a b n ≡
        ((Nat.digits p n).map fun d => centralTrinomial a b d).prod [ZMOD p]
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Proposition 4.
Uses 2
Statement usesGeneralized central trinomial coefficients · Laurent polynomials of degree at most one

TheoremDigit decomposition of shifted sums of constant terms modulo a prime#●ready to prove

Let \(\alpha, \beta, \gamma\) be integers with \(\alpha \ne 0\) or \(\gamma \ne 0\), let \(a_r\) be the constant term of \((\alpha x^{-1} + \beta + \gamma x)^r\), let \(p\) be a prime, let \(h < p\) be a natural number, let \(c_0, c_1, \ldots\) be integers, and let \(b_r = \sum_{i=0}^{h} c_i a_{r+i}\). Let \(q, k\) be natural numbers, let \(m, n_0 < p\) be natural numbers with \(m \ne p - 1\), put \(n = ((qp + m) p^k + p^k - 1) p + n_0\) (in base \(p\): the digits of \(q\), then \(m\), then \(k\) digits \(p - 1\), then \(n_0\)), and let \(C = \min(h, p - n_0 - 1)\). Then

\[b_n \equiv a_q \Big( a_m a_{p-1}^k \sum_{i=0}^{C} c_i a_{n_0+i} + a_{m+1} \sum_{i=p-n_0}^{h} c_i a_{i-(p-n_0)} \Big) \pmod p,\]

where the last sum is empty when \(h < p - n_0\).

public theorem_wanted constantTermLinearCombination_digit_decomposition
    (α β γ : ℤ) (hdegree : α ≠ 0 ∨ γ ≠ 0) (c : ℕ → ℤ)
    (h p q m k n₀ n : ℕ) (hp : p.Prime) (hph : h < p)
    (hm : m < p) (hn₀ : n₀ < p) (hmne : m ≠ p - 1)
    (hn : n = ((q * p + m) * p ^ k + (p ^ k - 1)) * p + n₀) :
    let a := fun r : ℕ => (linearLaurent α β γ ^ r).coeff 0
    let b := fun r : ℕ => ∑ i ∈ Finset.range (h + 1), c i * a (r + i)
    let C := min h (p - n₀ - 1)
    b n ≡ a q *
      (a m * a (p - 1) ^ k *
          (∑ i ∈ Finset.range (C + 1), c i * a (n₀ + i)) +
        a (m + 1) *
          (∑ i ∈ Finset.Icc (p - n₀) h, c i * a (i - (p - n₀))))
      [ZMOD p]
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Lemma 1.
Uses 1
Statement usesLaurent polynomials of degree at most one

TheoremDigit formula for shifted sums of constant terms, small last digit#●ready to prove

Let \(\alpha\), \(\beta\), \(\gamma\) be integers with \(\alpha \ne 0\) or \(\gamma \ne 0\), and let \(a_r\) be the constant term of \((\alpha x^{-1} + \beta + \gamma x)^r\). Let \(c_0, c_1, c_2, \dots\) be integers, let \(h\) be a natural number, and put \(b_r = \sum_{i=0}^{h} c_i a_{r+i}\). Let \(p\) be a prime with \(h < p\), and let \(q\), \(m\), \(k\), \(n_0\) be natural numbers with \(m < p\), \(m \ne p - 1\), \(n_0 < p\) and \(n_0 < p - h\). If \(n = ((qp + m)p^k + p^k - 1)p + n_0\), that is, \(n\) is obtained from \(q\) by appending the base-\(p\) digits \(m\), then \(k\) digits \(p - 1\), then \(n_0\), then

\[ b_n \equiv a_q \, a_m \, a_{p-1}^k \sum_{i=0}^{h} c_i a_{n_0+i} \pmod p \quad \text{and} \quad b_n \equiv a_{(n - n_0)/p} \, b_{n_0} \pmod p. \]
public theorem_wanted constantTermLinearCombination_small_last_digit
    (α β γ : ℤ) (hdegree : α ≠ 0 ∨ γ ≠ 0) (c : ℕ → ℤ)
    (h p q m k n₀ n : ℕ) (hp : p.Prime) (hph : h < p)
    (hm : m < p) (hn₀ : n₀ < p) (hmne : m ≠ p - 1)
    (hnsmall : n₀ < p - h)
    (hn : n = ((q * p + m) * p ^ k + (p ^ k - 1)) * p + n₀) :
    let a := fun r : ℕ => (linearLaurent α β γ ^ r).coeff 0
    let b := fun r : ℕ => ∑ i ∈ Finset.range (h + 1), c i * a (r + i)
    (b n ≡ a q * a m * a (p - 1) ^ k *
        (∑ i ∈ Finset.range (h + 1), c i * a (n₀ + i)) [ZMOD p]) ∧
      (b n ≡ a ((n - n₀) / p) * b n₀ [ZMOD p])
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Lemma 1.
Uses 1
Statement usesLaurent polynomials of degree at most one

TheoremDensity one for zeros of shifted combinations of central trinomial coefficients#●ready to prove

Let \(\alpha\) and \(\beta\) be integers with \(\alpha \neq 0\), and write \(a_r\) for the generalized central trinomial coefficient with parameters \(\alpha, \beta\), the constant term of \((\alpha x^{-1} + \beta + \alpha x)^r\). Let \(p\) be a prime, let \(h < p\) be a natural number, let \((c_i)_{i \ge 0}\) be a sequence of integers, and set \(b_r = \sum_{i=0}^{h} c_i a_{r+i}\) for \(r \in \mathbb{N}\). If \(a_r \equiv 0 \pmod p\) for some \(r \in \mathbb{N}\), then the set of \(r \in \mathbb{N}\) with \(b_r \equiv 0 \pmod p\) has natural density \(1\).

public theorem_wanted symmetricConstantTerm_zero_density_one
    (α β : ℤ) (hα : α ≠ 0) (c : ℕ → ℤ) (h p : ℕ)
    (hp : p.Prime) (hph : h < p)
    (hzero : ∃ r : ℕ, centralTrinomial α β r ≡ 0 [ZMOD p]) :
    let b := fun r : ℕ =>
      ∑ i ∈ Finset.range (h + 1), c i * centralTrinomial α β (r + i)
    ({r | b r ≡ 0 [ZMOD p]} : Set ℕ).HasDensity 1
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Lemma 2.
Uses 2
Statement usesGeneralized central trinomial coefficients · Density of a set

TheoremZero density of shifted combinations of central trinomial coefficients#●ready to prove

Let \(\alpha\) and \(\beta\) be integers with \(\alpha \neq 0\), and write \(a_r\) for the generalized central trinomial coefficient with parameters \(\alpha, \beta\), the constant term of \((\alpha x^{-1} + \beta + \alpha x)^r\). Let \(p\) be a prime, let \(h < p\) be a natural number, let \((c_i)_{i \ge 0}\) be a sequence of integers, and set \(b_r = \sum_{i=0}^{h} c_i a_{r+i}\) for \(r \in \mathbb{N}\). Suppose that \(a_r \not\equiv 0 \pmod p\) for every \(r \in \mathbb{N}\). For \(p - h \le r \le p - 1\) put \(C_r = \min(h, p - r - 1)\), \(S_r = \sum_{i=0}^{C_r} c_i a_{r+i}\) and \(S'_r = \sum_{i=C_r+1}^{h} c_i a_{i-(p-r)}\). Let \(N_0\) be the number of \(r\) with \(0 \le r < p - h\) and \(b_r \equiv 0 \pmod p\), let \(N_1\) be the number of pairs \((r, m)\) with \(p - h \le r \le p - 1\) and \(0 \le m \le p - 2\) such that \(a_m S_r \equiv -a_{m+1} S'_r \pmod p\), and let \(N_2\) be the number of pairs \((r, m)\) in that range such that \(a_m S_r \equiv -a_{p-1} a_{m+1} S'_r \pmod p\). Then the set of \(r \in \mathbb{N}\) with \(b_r \equiv 0 \pmod p\) has natural density

\[\frac{N_0}{p} + \frac{N_1}{(p-1)(p+1)} + \frac{N_2}{(p-1)p(p+1)}.\]

No assumption \(p > 2\) is made.

MetaMathlibExt.symmetricConstantTerm_zero_density_formula · 27 lines 
public theorem_wanted symmetricConstantTerm_zero_density_formula
    (α β : ℤ) (hα : α ≠ 0) (c : ℕ → ℤ) (h p : ℕ)
    (hp : p.Prime) (hph : h < p)
    (hnonzero : ∀ r : ℕ, ¬(centralTrinomial α β r ≡ 0 [ZMOD p])) :
    let a := centralTrinomial α β
    let b := fun r : ℕ => ∑ i ∈ Finset.range (h + 1), c i * a (r + i)
    let boundary := (Finset.Icc (p - h) (p - 1)).product (Finset.range (p - 1))
    let firstPairs := boundary.filter (fun nm =>
      let r := nm.1
      let m := nm.2
      let C := min h (p - r - 1)
      a m * (∑ i ∈ Finset.range (C + 1), c i * a (r + i)) ≡
        -(a (m + 1) * (∑ i ∈ Finset.Icc (C + 1) h,
          c i * a (i - (p - r)))) [ZMOD p])
    let secondPairs := boundary.filter (fun nm =>
      let r := nm.1
      let m := nm.2
      let C := min h (p - r - 1)
      a m * (∑ i ∈ Finset.range (C + 1), c i * a (r + i)) ≡
        -(a (p - 1) * a (m + 1) *
          (∑ i ∈ Finset.Icc (C + 1) h, c i * a (i - (p - r)))) [ZMOD p])
    ({r | b r ≡ 0 [ZMOD p]} : Set ℕ).HasDensity
      ((((Finset.range (p - h)).filter (fun r => b r ≡ 0 [ZMOD p])).card : ℝ) /
          (p : ℝ) +
        (firstPairs.card : ℝ) / (((p - 1 : ℕ) : ℝ) * (p + 1 : ℝ)) +
        (secondPairs.card : ℝ) /
          (((p - 1 : ℕ) : ℝ) * (p : ℝ) * (p + 1 : ℝ)))
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Lemma 2.
Uses 2
Statement usesGeneralized central trinomial coefficients · Density of a set

TheoremGeneralized Motzkin numbers modulo \(p\) with last digit below \(p - 2\)#●ready to prove

For integers \(a\), \(b\) let \(T_r\) and \(M_r\) be the constant terms of \((a x^{-1} + b + a x)^r\) and of \((a x^{-1} + b + a x)^r (1 - x^2)\), the generalized central trinomial coefficients and generalized Motzkin numbers. Let \(p > 2\) be a prime with \(b^2 - 4a^2 \not\equiv 0 \pmod p\), and let \(q\), \(n_0\) be natural numbers with \(n_0 < p - 2\). If \(n = qp + n_0\), then

\[ 2a^2 M_n \equiv 2a^2 \, T_q \, M_{n_0} \pmod p. \]
public theorem_wanted generalizedMotzkin_digit_below_prime_minus_two
    (a b : ℤ) (p q n₀ n : ℕ) (hp : p.Prime) (hp2 : 2 < p)
    (hdisc : ¬(b ^ 2 - 4 * a ^ 2 ≡ 0 [ZMOD p]))
    (hn₀ : n₀ < p - 2) (hn : n = q * p + n₀) :
    2 * a ^ 2 * generalizedMotzkin a b n ≡
      2 * a ^ 2 * centralTrinomial a b q * generalizedMotzkin a b n₀
      [ZMOD p]
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Proposition 5(i).
Uses 2
Statement usesGeneralized central trinomial coefficients · Generalized Motzkin numbers

TheoremGeneralized Motzkin numbers modulo \(p\) with last digit \(p - 2\)#●ready to prove

For integers \(a\), \(b\) let \(T_r\) and \(M_r\) be the constant terms of \((a x^{-1} + b + a x)^r\) and of \((a x^{-1} + b + a x)^r (1 - x^2)\), the generalized central trinomial coefficients and generalized Motzkin numbers. Let \(p > 2\) be a prime with \(b^2 - 4a^2 \not\equiv 0 \pmod p\), and let \(q\), \(m\), \(k\) be natural numbers with \(m < p - 1\). If \(n = ((qp + m)p^k + p^k - 1)p + p - 2\), that is, \(n\) is obtained from \(q\) by appending the base-\(p\) digits \(m\), then \(k\) digits \(p - 1\), then \(p - 2\), then

\[ 2a^2 M_n \equiv T_q \left( b \, T_m \, T_{p-1}^{k+1} - T_{m+1} \right) \pmod p. \]
public theorem_wanted generalizedMotzkin_digit_prime_minus_two
    (a b : ℤ) (p q m k n : ℕ) (hp : p.Prime) (hp2 : 2 < p)
    (hdisc : ¬(b ^ 2 - 4 * a ^ 2 ≡ 0 [ZMOD p])) (hm : m < p - 1)
    (hn : n = ((q * p + m) * p ^ k + (p ^ k - 1)) * p + (p - 2)) :
    2 * a ^ 2 * generalizedMotzkin a b n ≡
      centralTrinomial a b q *
        (b * centralTrinomial a b m * centralTrinomial a b (p - 1) ^ (k + 1) -
          centralTrinomial a b (m + 1))
      [ZMOD p]
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Proposition 5(ii).
Uses 2
Statement usesGeneralized central trinomial coefficients · Generalized Motzkin numbers

TheoremGeneralized Motzkin numbers modulo \(p\) with last digit \(p - 1\)#●ready to prove

For integers \(a\), \(b\) let \(T_r\) and \(M_r\) be the constant terms of \((a x^{-1} + b + a x)^r\) and of \((a x^{-1} + b + a x)^r (1 - x^2)\), the generalized central trinomial coefficients and generalized Motzkin numbers. Let \(p > 2\) be a prime with \(D = b^2 - 4a^2 \not\equiv 0 \pmod p\), let \(q\), \(m\), \(k\) be natural numbers with \(m < p - 1\), and put \(\ell = p - 2 - m\). If \(n = ((qp + m)p^k + p^k - 1)p + p - 1\), that is, \(n\) is obtained from \(q\) by appending the base-\(p\) digits \(m\), then \(k + 1\) digits \(p - 1\), then

\[ D^{(p-1)/2} \cdot 2a^2 M_n \equiv D^{m+1} \, T_q \left( b \, T_\ell - T_{p-1}^{k+1} \, T_{\ell+1} \right) \pmod p. \]

The source states this congruence with a power of the unit \(D\) whose exponent may be negative; here it is multiplied out so that every exponent is nonnegative.

public theorem_wanted generalizedMotzkin_digit_prime_minus_one
    (a b : ℤ) (p q m k n : ℕ) (hp : p.Prime) (hp2 : 2 < p)
    (hdisc : ¬(b ^ 2 - 4 * a ^ 2 ≡ 0 [ZMOD p])) (hm : m < p - 1)
    (hn : n = ((q * p + m) * p ^ k + (p ^ k - 1)) * p + (p - 1)) :
    let ℓ := p - 2 - m
    (b ^ 2 - 4 * a ^ 2) ^ ((p - 1) / 2) *
        (2 * a ^ 2 * generalizedMotzkin a b n) ≡
      (b ^ 2 - 4 * a ^ 2) ^ (m + 1) * centralTrinomial a b q *
        (b * centralTrinomial a b ℓ -
          centralTrinomial a b (p - 1) ^ (k + 1) *
            centralTrinomial a b (ℓ + 1))
      [ZMOD p]
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Proposition 5(iii).
Uses 2
Statement usesGeneralized central trinomial coefficients · Generalized Motzkin numbers

TheoremZeros of generalized Motzkin numbers with a small last digit#●ready to prove

Let \(a, b\) be integers and let \(p > 2\) be a prime with \(b^2 - 4a^2 \not\equiv 0 \pmod p\), and suppose that the generalized central trinomial coefficients satisfy \(T^{(a,b)}_r \not\equiv 0 \pmod p\) for every \(r < p\). Let \(q, n_0\) be natural numbers with \(n_0 < p - 2\) and let \(n = qp + n_0\). Then \(p\) divides the generalized Motzkin number \(M^{(a,b)}_n\) if and only if \(p\) divides \(M^{(a,b)}_{n_0}\).

public theorem_wanted generalizedMotzkin_zero_digit_below_prime_minus_two
    (a b : ℤ) (p q n₀ n : ℕ) (hp : p.Prime) (hp2 : 2 < p)
    (hdisc : ¬(b ^ 2 - 4 * a ^ 2 ≡ 0 [ZMOD p]))
    (hT : ∀ r < p, ¬(centralTrinomial a b r ≡ 0 [ZMOD p]))
    (hn₀ : n₀ < p - 2) (hn : n = q * p + n₀) :
    generalizedMotzkin a b n ≡ 0 [ZMOD p] ↔
      generalizedMotzkin a b n₀ ≡ 0 [ZMOD p]
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Corollary 3.
Uses 2
Statement usesGeneralized central trinomial coefficients · Generalized Motzkin numbers

TheoremZeros of generalized Motzkin numbers with last digit \(p - 2\)#●ready to prove

Let \(a, b\) be integers and let \(p > 2\) be a prime with \(a \not\equiv 0 \pmod p\) and \(b^2 - 4a^2 \not\equiv 0 \pmod p\). Write \(T_r = T^{(a,b)}_r\) for the generalized central trinomial coefficients and suppose that \(T_r \not\equiv 0 \pmod p\) for every \(r < p\). Let \(q, m, k\) be natural numbers with \(m < p - 1\) and let

\[n = \bigl((qp + m)p^k + p^k - 1\bigr)p + p - 2.\]

Then \(p\) divides the generalized Motzkin number \(M^{(a,b)}_n\) if and only if \(b T_m \equiv T_{p-1}^{k+1} T_{m+1} \pmod p\).

public theorem_wanted generalizedMotzkin_zero_digit_prime_minus_two
    (a b : ℤ) (p q m k n : ℕ) (hp : p.Prime) (hp2 : 2 < p)
    (ha : ¬(a ≡ 0 [ZMOD p]))
    (hdisc : ¬(b ^ 2 - 4 * a ^ 2 ≡ 0 [ZMOD p]))
    (hT : ∀ r < p, ¬(centralTrinomial a b r ≡ 0 [ZMOD p]))
    (hm : m < p - 1)
    (hn : n = ((q * p + m) * p ^ k + (p ^ k - 1)) * p + (p - 2)) :
    generalizedMotzkin a b n ≡ 0 [ZMOD p] ↔
      b * centralTrinomial a b m ≡
        centralTrinomial a b (p - 1) ^ (k + 1) *
          centralTrinomial a b (m + 1) [ZMOD p]
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Corollary 3. The hypothesis \(a \not\equiv 0 \pmod p\) is implicit in the paper, which cancels \(2a^2\).
Uses 2
Statement usesGeneralized central trinomial coefficients · Generalized Motzkin numbers

TheoremZeros of generalized Motzkin numbers with last digit \(p - 1\)#●ready to prove

Let \(a, b\) be integers and let \(p > 2\) be a prime with \(a \not\equiv 0 \pmod p\) and \(b^2 - 4a^2 \not\equiv 0 \pmod p\). Write \(T_r = T^{(a,b)}_r\) for the generalized central trinomial coefficients and suppose that \(T_r \not\equiv 0 \pmod p\) for every \(r < p\). Let \(q, m, k\) be natural numbers with \(m < p - 1\), let

\[n = \bigl((qp + m)p^k + p^k - 1\bigr)p + p - 1,\]

and let \(\ell = p - 2 - m\). Then \(p\) divides the generalized Motzkin number \(M^{(a,b)}_n\) if and only if \(b T_\ell \equiv T_{p-1}^{k+1} T_{\ell+1} \pmod p\).

public theorem_wanted generalizedMotzkin_zero_digit_prime_minus_one
    (a b : ℤ) (p q m k n : ℕ) (hp : p.Prime) (hp2 : 2 < p)
    (ha : ¬(a ≡ 0 [ZMOD p]))
    (hdisc : ¬(b ^ 2 - 4 * a ^ 2 ≡ 0 [ZMOD p]))
    (hT : ∀ r < p, ¬(centralTrinomial a b r ≡ 0 [ZMOD p]))
    (hm : m < p - 1)
    (hn : n = ((q * p + m) * p ^ k + (p ^ k - 1)) * p + (p - 1)) :
    let ℓ := p - 2 - m
    generalizedMotzkin a b n ≡ 0 [ZMOD p] ↔
      b * centralTrinomial a b ℓ ≡
        centralTrinomial a b (p - 1) ^ (k + 1) *
          centralTrinomial a b (ℓ + 1) [ZMOD p]
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Corollary 3. The hypothesis \(a \not\equiv 0 \pmod p\) is implicit in the paper, which cancels \(2a^2\).
Uses 2
Statement usesGeneralized central trinomial coefficients · Generalized Motzkin numbers

TheoremGeneralized Motzkin numbers vanish modulo a prime with density one#●ready to prove

Let \(a, b\) be integers and let \(p\) be a prime. If \(p\) divides the generalized central trinomial coefficient \(T^{(a,b)}_r\) for some \(r < p\), then the set of \(n \in \mathbb{N}\) such that \(p\) divides the generalized Motzkin number \(M^{(a,b)}_n\) has natural density \(1\).

public theorem_wanted generalizedMotzkin_zero_density_one
    (a b : ℤ) (p : ℕ) (hp : p.Prime)
    (hzero : ∃ r < p, centralTrinomial a b r ≡ 0 [ZMOD p]) :
    ({n | generalizedMotzkin a b n ≡ 0 [ZMOD p]} : Set ℕ).HasDensity 1
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Theorem 3.
Uses 3
Statement usesGeneralized central trinomial coefficients · Generalized Motzkin numbers · Density of a set

TheoremZero density formula for generalized Motzkin numbers modulo a prime#●ready to prove

Let \(a, b\) be integers and let \(p > 2\) be a prime with \(a \not\equiv 0 \pmod p\). Write \(T_r = T^{(a,b)}_r\) for the generalized central trinomial coefficients and \(M_n = M^{(a,b)}_n\) for the generalized Motzkin numbers, and suppose that \(T_r \not\equiv 0 \pmod p\) for every \(r < p\). Let \(A\) be the number of \(r < p - 2\) with \(M_r \equiv 0 \pmod p\), let \(B\) be the number of \(m < p - 1\) with \(b T_m \equiv T_{p-1} T_{m+1} \pmod p\), and let \(C\) be the number of \(m < p - 1\) with \(b T_m \equiv T_{m+1} \pmod p\). Then the set of \(n \in \mathbb{N}\) with \(M_n \equiv 0 \pmod p\) has natural density

\[D_0 = \frac{A}{p} + \frac{2B}{(p-1)(p+1)} + \frac{2C}{(p-1)p(p+1)}.\]
MetaMathlibExt.generalizedMotzkin_zero_density_formula · 18 lines 
public theorem_wanted generalizedMotzkin_zero_density_formula
    (a b : ℤ) (p : ℕ) (hp : p.Prime) (hp2 : 2 < p)
    (ha : ¬(a ≡ 0 [ZMOD p]))
    (hnonzero : ∀ r < p, ¬(centralTrinomial a b r ≡ 0 [ZMOD p])) :
    let D₀ :=
      ((((Finset.range (p - 2)).filter
          (fun r => generalizedMotzkin a b r ≡ 0 [ZMOD p])).card : ℝ) /
          (p : ℝ) +
        2 * (((Finset.range (p - 1)).filter (fun m =>
          b * centralTrinomial a b m ≡
            centralTrinomial a b (p - 1) * centralTrinomial a b (m + 1)
            [ZMOD p])).card : ℝ) /
          (((p - 1 : ℕ) : ℝ) * (p + 1 : ℝ)) +
        2 * (((Finset.range (p - 1)).filter (fun m =>
          b * centralTrinomial a b m ≡ centralTrinomial a b (m + 1)
            [ZMOD p])).card : ℝ) /
          (((p - 1 : ℕ) : ℝ) * (p : ℝ) * (p + 1 : ℝ)))
    ({n | generalizedMotzkin a b n ≡ 0 [ZMOD p]} : Set ℕ).HasDensity D₀
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Theorem 3. The hypothesis \(a \not\equiv 0 \pmod p\) is implicit in the paper, which cancels \(2a^2\).
Uses 3
Statement usesGeneralized central trinomial coefficients · Generalized Motzkin numbers · Density of a set

TheoremDensity of each nonzero residue among generalized Motzkin numbers modulo \(p\)#●ready to prove

For integers \(a\), \(b\) let \(T_r\) and \(M_r\) be the constant terms of \((a x^{-1} + b + a x)^r\) and of \((a x^{-1} + b + a x)^r (1 - x^2)\), the generalized central trinomial coefficients and generalized Motzkin numbers. Let \(p > 2\) be a prime with \(a \not\equiv 0 \pmod p\) such that \(T_r \not\equiv 0 \pmod p\) for every \(r < p\), and suppose that the residues of \(T_0, \dots, T_{p-1}\) generate the unit group \((\mathbb{Z}/p\mathbb{Z})^\times\). Put

\[ D_0 = \frac{N_1}{p} + \frac{2 N_2}{(p - 1)(p + 1)} + \frac{2 N_3}{(p - 1) p (p + 1)}, \]

where \(N_1\) is the number of \(r < p - 2\) with \(M_r \equiv 0 \pmod p\), \(N_2\) is the number of \(m < p - 1\) with \(b T_m \equiv T_{p-1} T_{m+1} \pmod p\), and \(N_3\) is the number of \(m < p - 1\) with \(b T_m \equiv T_{m+1} \pmod p\). Then for every nonzero residue \(z\) modulo \(p\), the set of natural numbers \(n\) with \(M_n \equiv z \pmod p\) has natural density \((1 - D_0)/(p - 1)\).

The hypothesis \(a \not\equiv 0 \pmod p\) is implicit in the source, which cancels \(2a^2\).

MetaMathlibExt.generalizedMotzkin_nonzero_density · 23 lines 
public theorem_wanted generalizedMotzkin_nonzero_density
    (a b : ℤ) (p : ℕ) (hp : p.Prime) (hp2 : 2 < p)
    (ha : ¬(a ≡ 0 [ZMOD p]))
    (hnonzero : ∀ r < p, ¬(centralTrinomial a b r ≡ 0 [ZMOD p]))
    (hgenerate : Subgroup.closure
      {u : (ZMod p)ˣ | ∃ r < p,
        (u : ZMod p) = (centralTrinomial a b r : ZMod p)} = ⊤)
    (z : ZMod p) (hz : z ≠ 0) :
    let D₀ :=
      ((((Finset.range (p - 2)).filter
          (fun r => generalizedMotzkin a b r ≡ 0 [ZMOD p])).card : ℝ) /
          (p : ℝ) +
        2 * (((Finset.range (p - 1)).filter (fun m =>
          b * centralTrinomial a b m ≡
            centralTrinomial a b (p - 1) * centralTrinomial a b (m + 1)
            [ZMOD p])).card : ℝ) /
          (((p - 1 : ℕ) : ℝ) * (p + 1 : ℝ)) +
        2 * (((Finset.range (p - 1)).filter (fun m =>
          b * centralTrinomial a b m ≡ centralTrinomial a b (m + 1)
            [ZMOD p])).card : ℝ) /
          (((p - 1 : ℕ) : ℝ) * (p : ℝ) * (p + 1 : ℝ)))
    ({n | (generalizedMotzkin a b n : ZMod p) = z} : Set ℕ).HasDensity
      ((1 - D₀) / ((p - 1 : ℕ) : ℝ))
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Theorem 3.
Uses 3
Statement usesGeneralized central trinomial coefficients · Generalized Motzkin numbers · Density of a set

TheoremZero density of Motzkin numbers modulo a prime#●ready to prove

Let \(p\) be a prime, and write \(M_n = M^{(1,1)}_n\) for the Motzkin numbers and \(T_r = T^{(1,1)}_r\) for the central trinomial coefficients. If \(T_r \equiv 0 \pmod p\) for some \(r < p\), then the set of \(n \in \mathbb{N}\) with \(M_n \equiv 0 \pmod p\) has natural density \(1\). If instead \(T_r \not\equiv 0 \pmod p\) for every \(r < p\) and \(p > 2\), let \(A\) be the number of \(r < p - 2\) with \(M_r \equiv 0 \pmod p\), let \(B\) be the number of \(m < p - 1\) with \(T_m \equiv T_{p-1} T_{m+1} \pmod p\), and let \(C\) be the number of \(m < p - 1\) with \(T_m \equiv T_{m+1} \pmod p\); then this set has natural density

\[D_0 = \frac{A}{p} + \frac{2B}{(p-1)(p+1)} + \frac{2C}{(p-1)p(p+1)},\]

and \(D_0 \ge \frac{2}{p(p-1)}\).

MetaMathlibExt.motzkin_zero_density · 20 lines 
public theorem_wanted motzkin_zero_density
    (p : ℕ) (hp : p.Prime) :
    ((∃ r < p, centralTrinomial 1 1 r ≡ 0 [ZMOD p]) →
      ({n | generalizedMotzkin 1 1 n ≡ 0 [ZMOD p]} : Set ℕ).HasDensity 1) ∧
    ((∀ r < p, ¬(centralTrinomial 1 1 r ≡ 0 [ZMOD p])) → 2 < p →
      let D₀ :=
        ((((Finset.range (p - 2)).filter
            (fun r => generalizedMotzkin 1 1 r ≡ 0 [ZMOD p])).card : ℝ) /
            (p : ℝ) +
          2 * (((Finset.range (p - 1)).filter (fun m =>
            centralTrinomial 1 1 m ≡
              centralTrinomial 1 1 (p - 1) * centralTrinomial 1 1 (m + 1)
              [ZMOD p])).card : ℝ) /
            (((p - 1 : ℕ) : ℝ) * (p + 1 : ℝ)) +
          2 * (((Finset.range (p - 1)).filter (fun m =>
            centralTrinomial 1 1 m ≡ centralTrinomial 1 1 (m + 1)
              [ZMOD p])).card : ℝ) /
            (((p - 1 : ℕ) : ℝ) * (p : ℝ) * (p + 1 : ℝ)))
      ({n | generalizedMotzkin 1 1 n ≡ 0 [ZMOD p]} : Set ℕ).HasDensity D₀ ∧
        2 / ((p : ℝ) * ((p - 1 : ℕ) : ℝ)) ≤ D₀)
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Corollary 4. The paper's conjecture-conditional middle clause is omitted.
Uses 3
Statement usesGeneralized central trinomial coefficients · Generalized Motzkin numbers · Density of a set

TheoremZero density of A005717 modulo a prime#●ready to prove

Let \(p\) be a prime, let \(T_m\) be the central trinomial coefficient, the constant term of \((x^{-1} + 1 + x)^m\), and let \(a_n\) be the constant term of \(x (x^{-1} + 1 + x)^n\) (OEIS A005717). If \(T_r \equiv 0 \pmod p\) for some \(r < p\), then the set of \(n \in \mathbb{N}\) with \(a_n \equiv 0 \pmod p\) has natural density \(1\). If instead \(T_r \not\equiv 0 \pmod p\) for every \(r < p\) and \(p > 2\), let \(N_1\) be the number of \(m < p - 1\) with \(T_m \equiv T_{p-1} T_{m+1} \pmod p\) and let \(N_2\) be the number of \(m < p - 1\) with \(T_m \equiv T_{m+1} \pmod p\). Then that set has natural density \(D_0 = (N_1 + p N_2) / ((p - 1)(p + 1))\), and \(D_0 \ge 1/(p - 1)\).

MetaMathlibExt.a005717_zero_density · 16 lines 
public theorem_wanted a005717_zero_density
    (p : ℕ) (hp : p.Prime) :
    ((∃ r < p, centralTrinomial 1 1 r ≡ 0 [ZMOD p]) →
      ({n | a005717 n ≡ 0 [ZMOD p]} : Set ℕ).HasDensity 1) ∧
    ((∀ r < p, ¬(centralTrinomial 1 1 r ≡ 0 [ZMOD p])) → 2 < p →
      let D₀ :=
        ((((Finset.range (p - 1)).filter (fun m =>
            centralTrinomial 1 1 m ≡
              centralTrinomial 1 1 (p - 1) * centralTrinomial 1 1 (m + 1)
              [ZMOD p])).card : ℝ) +
          p * (((Finset.range (p - 1)).filter (fun m =>
            centralTrinomial 1 1 m ≡ centralTrinomial 1 1 (m + 1)
              [ZMOD p])).card : ℝ)) /
          (((p - 1 : ℕ) : ℝ) * (p + 1 : ℝ))
      ({n | a005717 n ≡ 0 [ZMOD p]} : Set ℕ).HasDensity D₀ ∧
        1 / ((p - 1 : ℕ) : ℝ) ≤ D₀)
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Proposition 6.
Uses 3
Statement usesOEIS A005717 · Generalized central trinomial coefficients · Density of a set

TheoremZero density of Riordan numbers modulo a prime#●ready to prove

Let \(p\) be a prime, and write \(r_n\) for the Riordan numbers and \(T_m = T^{(1,1)}_m\) for the central trinomial coefficients. If \(T_m \equiv 0 \pmod p\) for some \(m < p\), then the set of \(n \in \mathbb{N}\) with \(r_n \equiv 0 \pmod p\) has natural density \(1\). If instead \(T_m \not\equiv 0 \pmod p\) for every \(m < p\) and \(p > 2\), let \(B\) be the number of \(m < p - 1\) with \(3 T_m \equiv T_{p-1} T_{m+1} \pmod p\) and let \(C\) be the number of \(m < p - 1\) with \(3 T_m \equiv T_{m+1} \pmod p\); then this set has natural density

\[D_0 = \frac{B + pC}{(p-1)(p+1)},\]

and \(D_0 \ge \frac{1}{p-1}\).

MetaMathlibExt.riordan_zero_density · 16 lines 
public theorem_wanted riordan_zero_density
    (p : ℕ) (hp : p.Prime) :
    ((∃ r < p, centralTrinomial 1 1 r ≡ 0 [ZMOD p]) →
      ({n | riordanNumber n ≡ 0 [ZMOD p]} : Set ℕ).HasDensity 1) ∧
    ((∀ r < p, ¬(centralTrinomial 1 1 r ≡ 0 [ZMOD p])) → 2 < p →
      let D₀ :=
        ((((Finset.range (p - 1)).filter (fun m =>
            3 * centralTrinomial 1 1 m ≡
              centralTrinomial 1 1 (p - 1) * centralTrinomial 1 1 (m + 1)
              [ZMOD p])).card : ℝ) +
          p * (((Finset.range (p - 1)).filter (fun m =>
            3 * centralTrinomial 1 1 m ≡ centralTrinomial 1 1 (m + 1)
              [ZMOD p])).card : ℝ)) /
          (((p - 1 : ℕ) : ℝ) * (p + 1 : ℝ))
      ({n | riordanNumber n ≡ 0 [ZMOD p]} : Set ℕ).HasDensity D₀ ∧
        1 / ((p - 1 : ℕ) : ℝ) ≤ D₀)
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Proposition 7.
Uses 3
Statement usesGeneralized central trinomial coefficients · Riordan numbers · Density of a set

TheoremZero density of A005773 modulo a prime#●ready to prove

Let \(p\) be a prime, let \(T_m\) be the central trinomial coefficient, the constant term of \((x^{-1} + 1 + x)^m\), and let \(a_n\) be the constant term of \((1 + x)(x^{-1} + 1 + x)^n\) (OEIS A005773). If \(T_r \equiv 0 \pmod p\) for some \(r < p\), then the set of \(n \in \mathbb{N}\) with \(a_n \equiv 0 \pmod p\) has natural density \(1\). If instead \(T_r \not\equiv 0 \pmod p\) for every \(r < p\) and \(p > 2\), let \(N_1\) be the number of \(m < p - 1\) with \(T_m \equiv -T_{p-1} T_{m+1} \pmod p\) and let \(N_2\) be the number of \(m < p - 1\) with \(T_m \equiv -T_{m+1} \pmod p\). Then that set has natural density \(D_0 = (N_1 + p N_2) / ((p - 1)(p + 1))\), and \(D_0 \ge 1/(p - 1)\).

MetaMathlibExt.a005773_zero_density · 16 lines 
public theorem_wanted a005773_zero_density
    (p : ℕ) (hp : p.Prime) :
    ((∃ r < p, centralTrinomial 1 1 r ≡ 0 [ZMOD p]) →
      ({n | a005773 n ≡ 0 [ZMOD p]} : Set ℕ).HasDensity 1) ∧
    ((∀ r < p, ¬(centralTrinomial 1 1 r ≡ 0 [ZMOD p])) → 2 < p →
      let D₀ :=
        ((((Finset.range (p - 1)).filter (fun m =>
            centralTrinomial 1 1 m ≡
              -(centralTrinomial 1 1 (p - 1) * centralTrinomial 1 1 (m + 1))
              [ZMOD p])).card : ℝ) +
          p * (((Finset.range (p - 1)).filter (fun m =>
            centralTrinomial 1 1 m ≡ -centralTrinomial 1 1 (m + 1)
              [ZMOD p])).card : ℝ)) /
          (((p - 1 : ℕ) : ℝ) * (p + 1 : ℝ))
      ({n | a005773 n ≡ 0 [ZMOD p]} : Set ℕ).HasDensity D₀ ∧
        1 / ((p - 1 : ℕ) : ℝ) ≤ D₀)
Sources
  • Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers Modulo p, INTEGERS 26 (2026), Article A22, Proposition 8.
Uses 3
Statement usesOEIS A005773 · Generalized central trinomial coefficients · Density of a set
9 fully proved25 ready to prove

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