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%I #15 Mar 14 2023 03:42:00
%S 1,1,9,166,4717,182136,8911549,528571408,36864033945,2956595372416,
%T 268116203622961,27128338649300736,3029974270053623941,
%U 370289278173654092800,49150116757136815109733,7041536364582774222616576,1083004122024520209576760369
%N E.g.f. satisfies A(x) = exp( 1/(1 - x * A(x)^3) - 1 ).
%H Winston de Greef, <a href="/A361094/b361094.txt">Table of n, a(n) for n = 0..322</a>
%F a(n) = n! * Sum_{k=0..n} (3*n+1)^(k-1) * binomial(n-1,n-k)/k!.
%F a(n) ~ (5 + sqrt(21))^n * n^(n-1) / (3^(3/4) * 7^(1/4) * 2^n * exp((3 - sqrt(21))/6 + (5 - sqrt(21))*n/2)). - _Vaclav Kotesovec_, Mar 02 2023
%t Table[n! * Sum[(3*n+1)^(k-1) * Binomial[n-1,n-k]/k!, {k,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Mar 02 2023 *)
%o (PARI) a(n) = n!*sum(k=0, n, (3*n+1)^(k-1)*binomial(n-1, n-k)/k!);
%Y Cf. A052873, A361093, A361095, A361096, A361097.
%Y Cf. A212917, A361066.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Mar 01 2023
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