bibkey: cigler2023catalanpowers authors: Johann Cigler year: 2023 title: “Some experimental observations about Hankel determinants of convolution powers of Catalan numbers” doi: 10.48550/arXiv.2308.07642 url: https://arxiv.org/abs/2308.07642v2 claim: “Experimentally conjectured closed forms for shifted Hankel determinants D_{r,s}(N) of the coefficients of c(x)^r, c the Catalan generating function; Section 2.2, Conjecture 11, gives D_{2k+1,m-k+1}((2k+1)n+k) for 0 ≤ m ≤ k+1.” strata_touched:

• D5/S3/Combinatorics/CatalanPowerHankel/CiglerEleven license: citation-only triage: anchor

Cigler, Hankel determinants of convolution powers of Catalan numbers

The paper studies the Hankel determinants D_{r,s}(N) = det(C_{r,i+j+s}){0 ≤ i,j < N} of the coefficients C{r,j} = r/(2j+r) · binom(2j+r, j) of the r-th power of the Catalan generating function, with C_{r,j} = 0 for negative j and D_{r,s}(0) = 1. It records computer experiments and states several closed forms as conjectures. In Section 2.2, Conjecture 11 asserts that for k ≥ 1, 0 ≤ m ≤ k+1 and n ≥ 0, D_{2k+1,m−k+1}((2k+1)n+k) = (−1)^{kn+binom(k,2)} (2k+1)^m (n+1)^m. The case m = 0 was proved in Cigler, arXiv:2403.11244, equation (22); the remaining cases were open.

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DOI: 10.48550/arXiv.2308.07642

URL: https://arxiv.org/abs/2308.07642v2

• Locator: Section 2.2, Conjecture 11, with definitions (1) and (2).