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%I #22 Nov 26 2021 09:43:53
%S 1,1,8,131,3303,113137,4909829,258275887,15974450676,1136164798581,
%T 91366516437475,8197719659916195,811910298234609913,
%U 87984131560596043801,10355660409438349522396,1315550191540192189444535,179413108433279983993509731
%N E.g.f. satisfies: log(A(x)) = exp(x*A(x)^3) - 1.
%H Seiichi Manyama, <a href="/A349683/b349683.txt">Table of n, a(n) for n = 0..328</a>
%F a(n) = Sum_{k=0..n} (3*n+1)^(k-1) * Stirling2(n,k).
%F 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
%t nterms=20;Table[Sum[(3n+1)^(k-1)*StirlingS2[n,k],{k,0,n}],{n,0,nterms-1}] (* _Paolo Xausa_, Nov 25 2021 *)
%o (PARI) a(n) = sum(k=0, n, (3*n+1)^(k-1)*stirling(n, k, 2));
%Y Cf. A349525, A349598.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Nov 25 2021
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