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%I #12 Apr 22 2024 03:34:19
%S 1,1,5,28,321,3636,65947,1154238,28622001,684987400,21513702771,
%T 656705784714,24936869827465,928288327257084,41315505985090443,
%U 1817727059210127286,92749773791662574433,4712674616532693996432,271157923143678988333027
%N E.g.f. A(x) satisfies A(x) = exp( x * (1+3*x)^(1/3) * A(x) ).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F E.g.f.: A(x) = exp( -LambertW(-x * (1+3*x)^(1/3)) ).
%F a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * 3^(n-k) * binomial(k/3,n-k)/k!.
%F a(n) ~ sqrt(4*r+1) * n^(n-1) / (exp(n - 5/2) * r^(n - 3/2)), where r = 0.29742497866288781360719311656731644994668261137281157848090655000... is the root of the equation r*(1 + 3*r)^(1/3) = exp(-1). - _Vaclav Kotesovec_, Apr 22 2024
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x*(1+3*x)^(1/3)))))
%o (PARI) a(n) = n!*sum(k=0, n, (k+1)^(k-1)*3^(n-k)*binomial(k/3, n-k)/k!);
%Y Cf. A362771, A372162.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Apr 20 2024
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