A377965 Expansion of e.g.f. (1+x)^2 * exp(x*(1+x)^2).

1, 3, 11, 55, 309, 1931, 13543, 101991, 828425, 7192819, 66002691, 639830423, 6510397501, 69266297595, 768989536799, 8876171274631, 106301772962193, 1318277355041891, 16892429768517115, 223330116792810999, 3041570471301007301, 42611228176879105003

Offset

0,2

Links

Table of n, a(n) for n=0..21.

Formula

a(n) = n! * Sum_{k=0..n} binomial(2*k+2,n-k) / k!.

From Vaclav Kotesovec, Nov 23 2024: (Start)

Recurrence: (n^2 - 3*n + 4)*a(n) = (n^2 - 3*n + 8)*a(n-1) + 2*(n-1)*(2*n^2 - 5*n + 4)*a(n-2) + 3*(n-2)*(n-1)*(n^2 - n + 2)*a(n-3).

a(n) ~ 3^(n/3 - 7/6) * exp(-4/81 + 3^(-7/3)*n^(1/3) + 2*3^(-2/3)*n^(2/3) - 2*n/3) * n^(2*(n+1)/3) * (1 + 5813*3^(1/3)/(4374*n^(1/3))). (End)

Prog

(PARI) a(n, s=2, t=2) = n!*sum(k=0, n, binomial(t*k+s, n-k)/k!);

Crossrefs

Cf. A377954, A377966.

Cf. A361278, A377963.

Cf. A343884.

Sequence in context: A242952 A266027 A306177 * A094259 A091845 A020061

Adjacent sequences: A377962 A377963 A377964 * A377966 A377967 A377968

Keyword

nonn,easy

Author

Seiichi Manyama, Nov 12 2024

Status

approved