A349683 E.g.f. satisfies: log(A(x)) = exp(x*A(x)^3) - 1.

1, 1, 8, 131, 3303, 113137, 4909829, 258275887, 15974450676, 1136164798581, 91366516437475, 8197719659916195, 811910298234609913, 87984131560596043801, 10355660409438349522396, 1315550191540192189444535, 179413108433279983993509731

Offset: 0

Author

Seiichi Manyama, Nov 25 2021

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..328

Crossrefs

Cf. A349525, A349598.

Programs

Mathematica

nterms=20;Table[Sum[(3n+1)^(k-1)*StirlingS2[n,k],{k,0,n}],{n,0,nterms-1}] (* Paolo Xausa, Nov 25 2021 *)

PARI

a(n) = sum(k=0, n, (3*n+1)^(k-1)*stirling(n, k, 2));

Formula

a(n) = Sum_{k=0..n} (3*n+1)^(k-1) * Stirling2(n,k).

a(n) ~ s * n^(n-1) / (3 * sqrt(1 + r*s^3) * exp(n) * r^n), where r = LambertW(1/3)/exp(1/LambertW(1/3) - 3) = 0.106691814639676411952403096776061... and s = exp(1/(3*LambertW(1/3)) - 1) = 1.341591995635184131204677967393502... are roots of the system of equations exp(r*s^3) = 1 + log(s), 3*r*s^3*exp(r*s^3) = 1. - Vaclav Kotesovec, Nov 26 2021