Whitt's generalized Catalan numbers

DefinitionWhitt's generalized Catalan numbers#✓fully proved

For a rational number \(\alpha\) and a natural number \(n\), Whitt's generalized Catalan number \(C(\alpha, n)\) is the rational number

\[C(\alpha, n) = \sum_{k=0}^{n-1} \left(1 - \frac{k}{n}\right) \binom{n-1+k}{k} \alpha^k.\]

In particular \(C(\alpha, 0) = 0\), the empty sum. Writing \(n + 1\) for \(n\) gives the source's form \(C(\alpha, n+1) = \sum_{k=0}^{n} \left(1 - \frac{k}{n+1}\right) \binom{n+k}{k} \alpha^k\). For \(\alpha = 2\) and \(n \ge 1\) these are the terms of OEIS A064062.

def whittGeneralizedCatalan (α : ℚ) (n : ℕ) : ℚ :=
  ∑ k ∈ Finset.range n, (1 - (k : ℚ) / n) * ((n - 1 + k).choose k : ℚ) * α ^ k
Sources
  • Sabar Taoufik, A New Triangle of Numbers: Properties and Observations, INTEGERS 26 (2026), Article A27, Section 5 (definition of C(α, n+1), quoting W. Whitt, J. Integer Seq. 14 (2011)).
Used by 8
Used byCentral entries of Sabar's triangle and Whitt's numbers · Fine numbers from Whitt's generalized Catalan numbers · Consecutive Whitt generalized Catalan numbers at \(\alpha = 2\) · Whitt's generalized Catalan number at one · Whitt's generalized Catalan numbers in the source's indexing · First values of Whitt's generalized Catalan numbers with parameter two · Whitt's generalized Catalan number at zero · Whitt's generalized Catalan numbers with parameter zero

TheoremWhitt's generalized Catalan numbers in the source's indexing#✓fully proved

For every rational number \(\alpha\) and every natural number \(n\), Whitt's generalized Catalan number \(C(\alpha, n + 1)\) is

\[C(\alpha, n + 1) = \sum_{k=0}^{n} \left(1 - \frac{k}{n+1}\right) \binom{n+k}{k} \alpha^k.\]
theorem whittGeneralizedCatalan_succ (α : ℚ) (n : ℕ) :
    whittGeneralizedCatalan α (n + 1) =
      ∑ k ∈ Finset.range (n + 1),
        (1 - (k : ℚ) / ((n : ℚ) + 1)) * ((n + k).choose k : ℚ) * α ^ k
Sources
  • Sabar Taoufik, A New Triangle of Numbers: Properties and Observations, INTEGERS 26 (2026), Article A27, Section 5 (definition of C(α, n+1)).
Uses 1
Statement usesWhitt's generalized Catalan numbers

TheoremWhitt's generalized Catalan number at zero#✓fully proved

For every rational number \(\alpha\), Whitt's generalized Catalan number \(C(\alpha, 0)\) equals \(0\), being an empty sum.

@[simp] theorem whittGeneralizedCatalan_zero (α : ℚ) : whittGeneralizedCatalan α 0 = 0
Sources
  • Sabar Taoufik, A New Triangle of Numbers: Properties and Observations, INTEGERS 26 (2026), Article A27, Section 5 (definition of C(α, n+1)).
Uses 1
Statement usesWhitt's generalized Catalan numbers

TheoremWhitt's generalized Catalan number at one#✓fully proved

For every rational number \(\alpha\), Whitt's generalized Catalan number \(C(\alpha, 1)\) equals \(1\).

@[simp] theorem whittGeneralizedCatalan_one (α : ℚ) : whittGeneralizedCatalan α 1 = 1
Sources
  • Sabar Taoufik, A New Triangle of Numbers: Properties and Observations, INTEGERS 26 (2026), Article A27, Section 5 (definition of C(α, n+1) at n = 0).
Uses 1
Statement usesWhitt's generalized Catalan numbers

TheoremWhitt's generalized Catalan numbers with parameter zero#✓fully proved

For every natural number \(n \ge 1\), Whitt's generalized Catalan number \(C(0, n)\) equals \(1\): only the term \(k = 0\) of the defining sum survives.

theorem whittGeneralizedCatalan_zero_left {n : ℕ} (hn : 0 < n) :
    whittGeneralizedCatalan 0 n = 1
Sources
  • Sabar Taoufik, A New Triangle of Numbers: Properties and Observations, INTEGERS 26 (2026), Article A27, Section 5 (definition of C(α, n+1)).
Uses 1
Statement usesWhitt's generalized Catalan numbers

TheoremFirst values of Whitt's generalized Catalan numbers with parameter two#✓fully proved

Whitt's generalized Catalan numbers with parameter \(\alpha = 2\) satisfy \(C(2, 1) = 1\), \(C(2, 2) = 3\), \(C(2, 3) = 13\) and \(C(2, 4) = 67\).

theorem whittGeneralizedCatalan_two_small :
    whittGeneralizedCatalan 2 1 = 1 ∧ whittGeneralizedCatalan 2 2 = 3 ∧
      whittGeneralizedCatalan 2 3 = 13 ∧ whittGeneralizedCatalan 2 4 = 67
Sources
  • Sabar Taoufik, A New Triangle of Numbers: Properties and Observations, INTEGERS 26 (2026), Article A27, Section 5 (OEIS A064062).
Uses 1
Statement usesWhitt's generalized Catalan numbers
6 fully proved

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