A381164 a(n) = Sum_{k=0..n} binomial(n,k)*(5*k)!/(k!)^5.

1, 121, 113641, 168508561, 306213587881, 624890127114721, 1374618918516663841, 3187068298971939367561, 7682172545187676630759081, 19079663136489248380982551201, 48525227073661262262248690661841, 125818607409307965748858681991235961, 331488456546076036761442657285875590881

Offset: 0

Author

Stefano Spezia, Feb 15 2025

Keywords

• nonn

Comments

Calabi-Yau series number 79.

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. A007318, A008978, A100734.

Cf. A381161, A381162, A381163, A381165.

Programs

Mathematica

a[n_]:=Sum[Binomial[n, k](5k)!/k!^5, {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-x))/(1-x).

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

a(n) == 1 (mod 120).

a(n) ~ 2^n * 3^(n+2) * 521^(n+2) / (5^(19/2) * Pi^2 * n^2). - Vaclav Kotesovec, May 29 2025