A381199 a(n) = (4*n)!/((n!)^2*(2*n)!)*Sum_{k=0..n} binomial(n,k)^2*binomial(2*k,k).

1, 36, 6300, 1718640, 575675100, 216636756336, 87874675224336, 37563969509352000, 16692217815436148700, 7642084994921759382000, 3582530520581922083974800, 1712083670316898167464884800, 831357643152788660610464490000, 409154554816583487288034143528000, 203690783136217174743485058666840000

Offset: 0

Author

Stefano Spezia, Feb 16 2025

Keywords

• nonn

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. 24.

Crossrefs

Cf. A000897, A000984, A001044, A002893, A007318, A008459, A010050, A100733.

Programs

Mathematica

a[n_]:=(4n)!/((n!)^2*(2n)!)*Sum[Binomial[n,k]^2Binomial[2k,k],{k,0,n}]; Array[a,15,0]

Formula

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

D-finite with recurrence n^4*a(n) -4*(4*n-1)*(4*n-3)*(10*n^2-10*n+3)*a(n-1) +144*(4*n-5)*(4*n-3)*(4*n-7)*(4*n-1)*a(n-2)=0. - R. J. Mathar, Feb 18 2025

a(n) ~ 2^(6*n - 1/2) * 3^(2*n + 3/2) / (4*Pi^2*n^2). - Vaclav Kotesovec, May 29 2025