Log-behavior of Motzkin prefix numbers

TheoremRatio bounds for the A005773 recurrence#●ready to prove

Let \((A_n)_{n \ge 0}\) be a sequence of real numbers with \(A_0 = A_1 = 1\) and \((n+1) A_{n+1} = 2(n+1) A_n + 3(n-1) A_{n-1}\) for every \(n \ge 1\). Write \(x_n = A_{n+1}/A_n\) and \(\varphi_n = 3(2n+1)/(2(n+1))\). Then for every \(n \ge 0\) we have \(x_n \le \varphi_n\) and

\[\frac{x_{n+1}}{x_n} \ge \frac{2(3n^2+6n+2)}{(n+2)(2n+1)\,\varphi_n}.\]

public theorem_wanted a005773Recurrence_ratio_bounds (A : ℕ → ℝ) (h0 : A 0 = 1) (h1 : A 1 = 1)
    (hrec : ∀ n : ℕ, 1 ≤ n →
      ((n : ℝ) + 1) * A (n + 1) = 2 * ((n : ℝ) + 1) * A n + 3 * ((n : ℝ) - 1) * A (n - 1))
    (n : ℕ) :
    A (n + 1) / A n ≤ 3 * (2 * n + 1) / (2 * (n + 1)) ∧
      2 * (3 * (n : ℝ) ^ 2 + 6 * n + 2) / ((n + 2) * (2 * n + 1) * (3 * (2 * n + 1) / (2 * (n + 1))))
        ≤ (A (n + 2) / A (n + 1)) / (A (n + 1) / A n)

Sources

• Feng-Zhen Zhao, The Log-Behavior of Some Sequences Related to a Linear Recurrence Sequence, INTEGERS 25 (2025), Article A100, Lemma 1.

TheoremThe first and second differences of A005773 are log-balanced#●ready to prove

Let \((A_n)_{n \ge 0}\) be a sequence of real numbers with \(A_0 = A_1 = 1\) and \((n+1) A_{n+1} = 2(n+1) A_n + 3(n-1) A_{n-1}\) for every \(n \ge 1\), and write \(\Delta A_n = A_{n+1} - A_n\) and \(\Delta^2 A_n = A_{n+2} - 2A_{n+1} + A_n\). Then for every \(n \ge 4\), each of the sequences \(z = \Delta A\) and \(z = \Delta^2 A\) satisfies \(z_n^2 \le z_{n-1} z_{n+1}\) and

\[\frac{z_{n-1}}{(n-1)!} \cdot \frac{z_{n+1}}{(n+1)!} \le \left(\frac{z_n}{n!}\right)^2.\]

That is, \((\Delta A_n)_{n \ge 3}\) and \((\Delta^2 A_n)_{n \ge 3}\) are log-balanced: log-convex, with \((z_n / n!)_{n \ge 3}\) log-concave.

public theorem_wanted a005773Recurrence_diff_logBalanced (A : ℕ → ℝ) (h0 : A 0 = 1)
    (h1 : A 1 = 1)
    (hrec : ∀ n : ℕ, 1 ≤ n →
      ((n : ℝ) + 1) * A (n + 1) = 2 * ((n : ℝ) + 1) * A n + 3 * ((n : ℝ) - 1) * A (n - 1))
    (n : ℕ) (hn : 4 ≤ n) :
    let D₁ : ℕ → ℝ := fun m => A (m + 1) - A m
    let D₂ : ℕ → ℝ := fun m => A (m + 2) - 2 * A (m + 1) + A m
    (D₁ n ^ 2 ≤ D₁ (n - 1) * D₁ (n + 1) ∧
        D₁ (n - 1) / (n - 1).factorial * (D₁ (n + 1) / (n + 1).factorial) ≤
          (D₁ n / n.factorial) ^ 2) ∧
      D₂ n ^ 2 ≤ D₂ (n - 1) * D₂ (n + 1) ∧
        D₂ (n - 1) / (n - 1).factorial * (D₂ (n + 1) / (n + 1).factorial) ≤
          (D₂ n / n.factorial) ^ 2

Sources

• Feng-Zhen Zhao, The Log-Behavior of Some Sequences Related to a Linear Recurrence Sequence, INTEGERS 25 (2025), Article A100, Theorem 1.

TheoremThe third differences of A005773 are log-convex and log-balanced#●ready to prove

Let \((A_n)_{n \ge 0}\) be a sequence of real numbers with \(A_0 = A_1 = 1\) and \((n+1) A_{n+1} = 2(n+1) A_n + 3(n-1) A_{n-1}\) for every \(n \ge 1\), and write \(z_n = \Delta^3 A_n = A_{n+3} - 3A_{n+2} + 3A_{n+1} - A_n\). Then \(z_n^2 \le z_{n-1} z_{n+1}\) for every \(n \ge 3\), and

\[\frac{z_{n-1}}{(n-1)!} \cdot \frac{z_{n+1}}{(n+1)!} \le \left(\frac{z_n}{n!}\right)^2\]

for every \(n \ge 4\). That is, \((\Delta^3 A_n)_{n \ge 2}\) is log-convex and \((z_n / n!)_{n \ge 3}\) is log-concave, so \((\Delta^3 A_n)_{n \ge 3}\) is log-balanced.

public theorem_wanted a005773Recurrence_thirdDiff_logConvex_logBalanced (A : ℕ → ℝ)
    (h0 : A 0 = 1) (h1 : A 1 = 1)
    (hrec : ∀ n : ℕ, 1 ≤ n →
      ((n : ℝ) + 1) * A (n + 1) = 2 * ((n : ℝ) + 1) * A n + 3 * ((n : ℝ) - 1) * A (n - 1)) :
    let D₃ : ℕ → ℝ := fun m => A (m + 3) - 3 * A (m + 2) + 3 * A (m + 1) - A m
    (∀ n, 3 ≤ n → D₃ n ^ 2 ≤ D₃ (n - 1) * D₃ (n + 1)) ∧
      ∀ n, 4 ≤ n →
        D₃ (n - 1) / (n - 1).factorial * (D₃ (n + 1) / (n + 1).factorial) ≤
          (D₃ n / n.factorial) ^ 2

Sources

• Feng-Zhen Zhao, The Log-Behavior of Some Sequences Related to a Linear Recurrence Sequence, INTEGERS 25 (2025), Article A100, Theorem 2.

TheoremLog-concave weighted sequences built from A005773#●ready to prove

Let \((A_n)_{n \ge 0}\) be a sequence of real numbers with \(A_0 = A_1 = 1\) and \((n+1) A_{n+1} = 2(n+1) A_n + 3(n-1) A_{n-1}\) for every \(n \ge 1\), and write \(\Delta A_n = A_{n+1} - A_n\) and \(\Delta^2 A_n = A_{n+2} - 2A_{n+1} + A_n\). Then for every \(n \ge 2\), each of the four sequences \(z_m = m A_m\), \(z_m = (m+1) A_{m+1} - m A_m\), \(z_m = m\,\Delta A_m\) and \(z_m = m\,\Delta^2 A_m\) satisfies \(z_{n-1} z_{n+1} \le z_n^2\). That is, these four sequences, indexed from \(m = 1\), are log-concave.

public theorem_wanted a005773Recurrence_weighted_logConcave (A : ℕ → ℝ) (h0 : A 0 = 1)
    (h1 : A 1 = 1)
    (hrec : ∀ n : ℕ, 1 ≤ n →
      ((n : ℝ) + 1) * A (n + 1) = 2 * ((n : ℝ) + 1) * A n + 3 * ((n : ℝ) - 1) * A (n - 1))
    (n : ℕ) (hn : 2 ≤ n) :
    let W : ℕ → ℝ := fun m => m * A m
    let V : ℕ → ℝ := fun m => (m + 1 : ℕ) * A (m + 1) - m * A m
    let U : ℕ → ℝ := fun m => m * (A (m + 1) - A m)
    let T : ℕ → ℝ := fun m => m * (A (m + 2) - 2 * A (m + 1) + A m)
    W (n - 1) * W (n + 1) ≤ W n ^ 2 ∧ V (n - 1) * V (n + 1) ≤ V n ^ 2 ∧
      U (n - 1) * U (n + 1) ≤ U n ^ 2 ∧ T (n - 1) * T (n + 1) ≤ T n ^ 2

Sources

• Feng-Zhen Zhao, The Log-Behavior of Some Sequences Related to a Linear Recurrence Sequence, INTEGERS 25 (2025), Article A100, Theorem 3.

TheoremLog-convexity of \((n A_n)^n\) for A005773#●ready to prove

Let \((A_n)_{n \ge 0}\) be a sequence of real numbers with \(A_0 = A_1 = 1\) and \((n+1) A_{n+1} = 2(n+1) A_n + 3(n-1) A_{n-1}\) for every \(n \ge 1\), and let \(z_m = (m A_m)^m\). Then \(z_n^2 \le z_{n-1} z_{n+1}\) for every \(n \ge 2\); that is, the sequence \((n^n A_n^n)_{n \ge 1}\) is log-convex.

public theorem_wanted a005773Recurrence_pow_logConvex (A : ℕ → ℝ) (h0 : A 0 = 1)
    (h1 : A 1 = 1)
    (hrec : ∀ n : ℕ, 1 ≤ n →
      ((n : ℝ) + 1) * A (n + 1) = 2 * ((n : ℝ) + 1) * A n + 3 * ((n : ℝ) - 1) * A (n - 1))
    (n : ℕ) (hn : 2 ≤ n) :
    let z : ℕ → ℝ := fun m => ((m : ℝ) * A m) ^ m
    z n ^ 2 ≤ z (n - 1) * z (n + 1)

Sources

• Feng-Zhen Zhao, The Log-Behavior of Some Sequences Related to a Linear Recurrence Sequence, INTEGERS 25 (2025), Article A100, Theorem 4.

5 ready to prove

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