A383378 Expansion of e.g.f. exp(-3*x) / (1-x)^4.

1, 1, 5, 21, 129, 897, 7317, 67365, 692577, 7849953, 97199109, 1304688789, 18863836065, 292198665249, 4826470920021, 84669407740773, 1571901715253313, 30786460730863425, 634323280633460613, 13714611211502376597, 310448651226154786881, 7342298348439393120321

Offset: 0

Author

Seiichi Manyama, Apr 24 2025

Keywords

• nonn
• easy

Crossrefs

Column k=3 of A383341.

Cf. A000261, A383344, A383380.

Cf. A010843, A137775, A383382.

Programs

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-3*x)/(1-x)^4))

Formula

a(n) = n! * Sum_{k=0..n} (-3)^(n-k) * binomial(k+3,3)/(n-k)!.

a(0) = a(1) = 1; a(n) = n*a(n-1) + 3*(n-1)*a(n-2).

a(n) = A137775(n+2)/(3*(n+1)).

a(n) ~ sqrt(2*Pi) * n^(n + 7/2) / (6*exp(n+3)). - Vaclav Kotesovec, Apr 25 2025