A353999 Expansion of e.g.f. 1/(1 - x^3/6 * (exp(x) - 1)).

1, 0, 0, 0, 4, 10, 20, 35, 1176, 10164, 58920, 277365, 3363580, 47567806, 519759604, 4591587455, 51017687280, 786120055400, 12187597925136, 165128862881769, 2261843835692340, 36940778814100210, 678763188831800380, 12143893591131411571, 211404290379223149384

Offset: 0

Author

Seiichi Manyama, May 13 2022

Keywords

• nonn

Crossrefs

Cf. A052848, A353998.

Cf. A000292, A351506, A354001.

Programs

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x^3/6*(exp(x)-1))))

PARI

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i!/6*sum(j=4, i, 1/(j-3)!*v[i-j+1]/(i-j)!)); v;

PARI

a(n) = n!*sum(k=0, n\4, k!*stirling(n-3*k, k, 2)/(6^k*(n-3*k)!));

Formula

a(0) = 1; a(n) = n!/6 * Sum_{k=4..n} 1/(k-3)! * a(n-k)/(n-k)! = binomial(n,3) * Sum_{k=4..n} binomial(n-3,k-3) * a(n-k).

a(n) = n! * Sum_{k=0..floor(n/4)} k! * Stirling2(n-3*k,k)/(6^k * (n-3*k)!).