A355408 Expansion of e.g.f. 1/(1 + exp(x) - exp(3*x)).

1, 2, 16, 170, 2416, 42962, 916696, 22819610, 649207456, 20778364322, 738918769576, 28905116527850, 1233506128752496, 57025618592932082, 2839117599033828856, 151446758367400488890, 8617182795217834505536, 520954229292164353554242

Offset: 0

Author

Seiichi Manyama, Jul 01 2022

Keywords

• nonn

Crossrefs

Cf. A000556, A355378, A355409.

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+exp(x)-exp(3*x))))

PARI

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (3^j-1)*binomial(i, j)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} (3^k - 1) * binomial(n,k) * a(n-k).

a(n) ~ n! / ((3 + 2*r) * log(r)^(n+1)), where r = 2*cosh(log((25 + 3*sqrt(69)) / 2) / 6)/sqrt(3). - Vaclav Kotesovec, Jul 01 2022