2026/07/28 by Dorin Dumitraşcu, Liviu Suciu
#math.CO
For a positive integer d≥ 1, we consider the sequences (An(d))n and (xn(d))n given by An(d) =∑n1+…+nd=n ((2n)!)/((n1!)2 (n2!)2 … (nd!)2) and xn(d) = \fracAn(d)\binom2nn. They have rich combinatorial interpretations, but we focus on the analytical properties of their generating functions Ad and Fd. We use a modified Borel transform, and algebraic and combinatorial considerations to prove that Fd is annihilated by an irreducible Fuchsian differentiable operator Ld-1,F of order d-1. We determine the structure of Fd as a global analytic function (analytic continuations from the original disk of definition, branches, finite singularities, and the structure of Fd near the finite singularities). Additionally, we show that the sequence (xn(d))n satisfies a minimal recurrence of width r=\lfloor (d+1)/2 \rfloor with polynomial coefficients Qr(n+r) xn+r+⋯ + Q0(n) xn=0, n ≥ 0. These polynomials are shown to have very specific symmetries and we compute explicitly Q0, Q1, and Qr. Similar results about the functions Ad are obtained.