Patalan numbers

DefinitionBinomial coefficient with rational upper argument#✓fully proved

For a rational number \(r\) and a natural number \(k\), the generalized binomial coefficient is the rational number \(\binom{r}{k} = r(r-1)\cdots(r-k+1)/k!\), which is \(1\) for \(k = 0\).

It is the binomial coefficient with fractional upper argument used in the \((p,q)\)-Patalan numbers.

def patalanGeneralizedBinomial (r : ℚ) (k : ℕ) : ℚ :=
  Finset.prod (Finset.range k) (fun i => r - (i : ℚ)) / (Nat.factorial k : ℚ)
Sources
Used by 2
Used byThe (p,q)-Patalan numbers · The (p,q)-super Patalan numbers

StructureAdmissible parameters for the (p,q)-Patalan numbers#✓fully proved

An admissible parameter pair for the \((p,q)\)-Patalan numbers is a pair of natural numbers \(p\) and \(q\) with \(p > 1\) and \(0 < q < p\).

structure PQPatalanParams where
  p : ℕ
  q : ℕ
  hp : 1 < p
  hq_pos : 0 < q
  hq : q < p
Sources
Used by 3
Used byThe (p,q)-Patalan numbers · The (p,q)-super Patalan numbers · Super Patalan numbers of order p

DefinitionThe (p,q)-Patalan numbers#✓fully proved

For natural numbers \(p\) and \(q\) with \(p > 1\) and \(0 < q < p\), the \((p,q)\)-Patalan number with index \(n \in \mathbb{N}\) is the rational number \(b(n) = -p^{2n+1} \binom{n - q/p}{n+1}\), where \(\binom{r}{k} = r(r-1)\cdots(r-k+1)/k!\) is the binomial coefficient with rational upper argument \(r\).

def pqPatalanNumber (P : PQPatalanParams) (n : ℕ) : ℚ :=
  -((P.p : ℚ) ^ (2 * n + 1)) *
    patalanGeneralizedBinomial ((n : ℚ) - (P.q : ℚ) / (P.p : ℚ)) (n + 1)
Sources
Uses 2
Statement usesAdmissible parameters for the (p,q)-Patalan numbers · Binomial coefficient with rational upper argument

DefinitionThe (p,q)-super Patalan numbers#✓fully proved

For natural numbers \(p\) and \(q\) with \(p > 1\) and \(0 < q < p\), and natural numbers \(i\) and \(j\), the \((p,q)\)-super Patalan number is the rational number \(Q(i,j) = (-1)^j p^{2(i+j)} \binom{i - q/p}{i+j}\), where \(\binom{r}{k} = r(r-1)\cdots(r-k+1)/k!\) is the binomial coefficient with rational upper argument \(r\).

def pqSuperPatalanNumber (P : PQPatalanParams) (i j : ℕ) : ℚ :=
  (-1) ^ j * (P.p : ℚ) ^ (2 * (i + j)) *
    patalanGeneralizedBinomial ((i : ℚ) - (P.q : ℚ) / P.p) (i + j)
Sources
  • Thomas M. Richardson, "The Super Patalan Numbers," Journal of Integer Sequences 18 (2015), Article 15.3.3.
Uses 2 · Used by 1
Statement usesAdmissible parameters for the (p,q)-Patalan numbers · Binomial coefficient with rational upper argument
Used bySuper Patalan numbers of order p

DefinitionSuper Patalan numbers of order p#✓fully proved

For a natural number \(p > 1\) and natural numbers \(i\) and \(j\), the super Patalan number of order \(p\) is the \((p,q)\)-super Patalan number with \(q = 1\), that is, the rational number \((-1)^j p^{2(i+j)} \binom{i - 1/p}{i+j}\), where \(\binom{r}{k} = r(r-1)\cdots(r-k+1)/k!\) is the binomial coefficient with rational upper argument \(r\).

def superPatalanNumber (p i j : ℕ) (hp : 1 < p) : ℚ :=
  pqSuperPatalanNumber
    { p := p, q := 1, hp := hp, hq_pos := by omega, hq := hp } i j
Uses 2
Statement usesAdmissible parameters for the (p,q)-Patalan numbers · The (p,q)-super Patalan numbers
5 fully proved

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