A381165 a(n) = Sum_{k=0..n} binomial(2*n,n)*binomial(n, k)*(5*k)!/((k!)^3*(2*k)!).

1, 122, 114126, 169305620, 307902541870, 628881704226972, 1384648756554128604, 3213280613371692112392, 7752574653184355259506670, 19272593072633780827550508620, 49062146831202726778631520779476, 127331178560917294198014376933764792, 335791906923524740189894975371277920796

Offset: 0

Author

Stefano Spezia, Feb 15 2025

Keywords

• nonn

Comments

Calabi-Yau series number 128.

Links

• S. Hassani, J.-M. Maillard, and N. Zenine, On the diagonals of rational functions: the minimal number of variables (unabridged version), arXiv:2502.05543 [math-ph], 2025. See p. 16.

Crossrefs

Cf. A000442, A000984, A007318, A010050, A100734.

Cf. A381161, A381162, A381163, A381164.

Programs

Mathematica

a[n_]:=Sum[Binomial[2n,n]Binomial[n, k](5k)!/((k!)^3*(2k)!), {k, 0, n}]; Array[a, 13, 0]

Formula

G.f.: hypergeom([1/5, 2/5, 3/5, 4/5], [1, 1, 1], 5^5*x/(1-4*x))/sqrt(1-4*x).

a(n) = binomial(2*n,n)*hypergeom([1/5, 2/5, 3/5, 4/5, -n], [1/2, 1, 1, 1], -5^5/4).

a(n) ~ 3^(n + 3/2) * 7^(n + 3/2) * 149^(n +3/2) / (4 * 5^7 * Pi^2 * n^2). - Vaclav Kotesovec, May 29 2025