A355410 Expansion of e.g.f. 1/(3 - exp(x) - exp(3*x)).

1, 4, 42, 652, 13482, 348484, 10809282, 391162972, 16177467642, 752689508404, 38911563009522, 2212759299753292, 137270821971529002, 9225382887659221924, 667690580181890112162, 51776098497454677943612, 4282645413209764715753562

Offset: 0

Author

Seiichi Manyama, Jul 01 2022

Keywords

• nonn

Crossrefs

Cf. A004700, A355408.

Programs

Maple

A355410 := proc(n)
    option remember ;
    if n = 0 then
        1;
    else
        add((3^k + 1) * binomial(n,k) * procname(n-k),k=1..n) ;
    end if;
end proc:
seq(A355410(n),n=0..70) ; # R. J. Mathar, Dec 04 2023

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(3-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! / ((9 - 2*r) * log(r)^(n+1)), where r = -2*sinh(log((-9*sqrt(3) + sqrt(247))/2)/3)/sqrt(3). - Vaclav Kotesovec, Jul 01 2022