A097821 Expansion of e.g.f. exp(2x)/(1-5x).

1, 7, 74, 1118, 22376, 559432, 16783024, 587405968, 23496238976, 1057330754432, 52866537722624, 2907659574746368, 174459574484786176, 11339872341511109632, 793791063905777690624, 59534329792933326829568

Offset: 0

Author

Paul Barry, Aug 26 2004

Keywords

• easy
• nonn

Comments

Second binomial transform of n!*5^n.

Links

• Robert Israel, Table of n, a(n) for n = 0..353

Crossrefs

Cf. A056545, A000354.

Programs

Maple

f:= proc(n) option remember; 5*n*procname(n-1)+2^n end proc:
f(0):= 1:
map(f, [$0..50]); # Robert Israel, Nov 10 2022

PARI

my(x='x + O('x^25)); Vec(serlaplace(exp(2*x)/(1-5*x))) \\ Michel Marcus, Nov 08 2022

Formula

a(n) = 5*n*a(n-1) + 2^n, n > 0, a(0)=1.

D-finite with recurrence a(n) +(-5*n-2)*a(n-1) +10*(n-1)*a(n-2)=0. - R. J. Mathar, Aug 20 2021

a(n) = 5^n * n! * Sum_{k = 0..n} (2/5)^k/k! = 5^n * exp(2/5) * gamma(n + 1, 2/5). - Gerry Martens, Nov 07 2022