A381163 a(n) = Sum_{k=0..n} binomial(n,k)*(4*k)!*(2*k)!/(k!)^6.
1, 49, 15217, 7437505, 4444068913, 2978797867489, 2151085262277121, 1636678166183569873, 1294384621280668799665, 1054623536679756097536097, 879831837105310233485202337, 748258333337818719124808979313, 646586399881218539235007860940609, 566284969531710881501724274920081265
Offset: 0
Keywords
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 pp. 15-16.
Programs
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Mathematica
a[n_]:=Sum[Binomial[n,k](4k)!(2k)!/k!^6,{k,0,n}]; Array[a,14,0]
Formula
G.f.: hypergeom([1/2, 1/2, 1/4, 3/4], [1, 1, 1], 2^10*x/(1-x))/(1-x).
a(n) = hypergeom([1/4, 1/2, 1/2, 3/4, -n], [1, 1, 1, 1], -2^10).
a(n) == 1 (mod 48).
a(n) ~ 5^(2*n+4) * 41^(n+2) / (2^(41/2) * Pi^2 * n^2). - Vaclav Kotesovec, May 29 2025
Comments