A371814 a(n) = Sum_{k=0..n} (-1)^k * binomial(4*n-k-1,n-k).

1, 2, 16, 128, 1068, 9142, 79612, 701864, 6244892, 55962920, 504375396, 4567003520, 41513817444, 378596616452, 3462411408136, 31742042431048, 291616814436124, 2684123914512280, 24746511514749280, 228491677484832896, 2112549277665243328

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Formula

a(n) = [x^n] 1/((1+x) * (1-x)^(3*n)).

a(n) = binomial(4*n-1, n)*hypergeom([1, -n], [1-4*n], -1). - Stefano Spezia, Apr 07 2024

From Vaclav Kotesovec, Apr 07 2024: (Start)

Recurrence: 24*n*(3*n - 2)*(3*n - 1)*(415*n^3 - 1898*n^2 + 2871*n - 1436)*a(n) = (838715*n^6 - 5099533*n^5 + 12225995*n^4 - 14652035*n^3 + 9157250*n^2 - 2799192*n + 322560)*a(n-1) + 8*(2*n - 3)*(4*n - 7)*(4*n - 5)*(415*n^3 - 653*n^2 + 320*n - 48)*a(n-2).

a(n) ~ 2^(8*n + 1/2) / (5 * sqrt(Pi*n) * 3^(3*n - 1/2)). (End)

Prog

(PARI) a(n) = sum(k=0, n, (-1)^k*binomial(4*n-k-1, n-k));

Crossrefs

Cf. A072547, A371813.

Cf. A005810, A262977.

Sequence in context: A338933 A013730 A270924 * A214623 A333719 A161737

Adjacent sequences: A371811 A371812 A371813 * A371815 A371816 A371817

Keyword

nonn

Author

Seiichi Manyama, Apr 06 2024

Status

approved