A368324 Expansion of e.g.f. exp(4*x) / (4 - 3*exp(x)).

1, 7, 61, 679, 9589, 167167, 3488221, 84882679, 2360477509, 73846507567, 2566948755181, 98151533290279, 4094166835331029, 185010377629273567, 9003499122285420541, 469450479424436219479, 26109404756356597154149, 1542883354097286642881167

Offset: 0

Author

Seiichi Manyama, Dec 21 2023

Keywords

• nonn

Crossrefs

Cf. A032033, A201354, A368322, A368323.

Programs

PARI

b(n, t) = sum(k=0, n, t^k*k!*stirling(n, k, 2));
a(n, m=4, t=3) = my(u=1+1/t); u^m*b(n, t)-(1/t)*sum(j=0, m-1, u^j*(m-1-j)^n);

Formula

a(n) = 4^n + 3 * Sum_{k=1..n} binomial(n,k) * a(n-k).

a(n) = (256/81)*A032033(n) - (1/3)*(3^n + (4/3)*2^n + 16/9 + (64/27)*0^n).