A351493 Expansion of e.g.f. (1 - x)^(-x^3/6).

1, 0, 0, 0, 4, 10, 40, 210, 1904, 15120, 132600, 1293600, 14303520, 171531360, 2223464880, 31023392400, 464541960960, 7424367350400, 126124766476800, 2269425252931200, 43119553374460800, 862673918061715200, 18126931548822835200, 399119899456951411200

Offset: 0

Author

Seiichi Manyama, May 02 2022

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..451

Crossrefs

Cf. A351492, A351506, A353229.

Programs

PARI

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

PARI

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

PARI

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

PARI

a(n) = n!*sum(k=0, n\4, abs(stirling(n-3*k, k, 1))/(6^k*(n-3*k)!));

Formula

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

a(n) = n! * Sum_{k=0..floor(n/4)} |Stirling1(n-3*k,k)|/(6^k * (n-3*k)!).

a(n) ~ sqrt(2*Pi) * n^(n - 1/3) / (Gamma(1/6) * exp(n)). - Vaclav Kotesovec, May 04 2022