A383381 Expansion of e.g.f. exp(-2*x) / (1-x)^5.

1, 3, 14, 82, 576, 4688, 43264, 445632, 5062016, 62812288, 844863744, 12239474432, 189939644416, 3142842052608, 55223903596544, 1026805938614272, 20139224002953216, 415503046091767808, 8994794537935765504, 203848794955954716672, 4826475681472562855936, 119162892472107134353408

Offset: 0

Author

Seiichi Manyama, Apr 24 2025

Keywords

• nonn
• easy

Crossrefs

Cf. A001909, A383382, A383383, A383384.

Cf. A000023, A052124, A087981, A383380.

Programs

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-2*x)/(1-x)^5))

Formula

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

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

a(n) ~ sqrt(2*Pi) * n^(n + 9/2) / (24*exp(n+2)). - Vaclav Kotesovec, Apr 25 2025