A365053 E.g.f. satisfies A(x) = exp( x * (1+x/2) * A(x) ).
1, 1, 4, 25, 230, 2786, 42112, 764296, 16209916, 393678856, 10777609556, 328466815964, 11031378197776, 404830360798072, 16118917055902312, 692126238230304616, 31882272572881781648, 1568365865590875789824, 82061348851406564851312
Offset: 0
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..379
- Eric Weisstein's World of Mathematics, Lambert W-Function.
Programs
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PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x*(1+x/2)))))
Formula
E.g.f.: exp( -LambertW(-x * (1+x/2)) ).
a(n) = n! * Sum_{k=0..n} (1/2)^(n-k) * (k+1)^(k-1) * binomial(k,n-k)/k!.
From Vaclav Kotesovec, Nov 10 2023: (Start)
E.g.f.: -LambertW(-x * (1+x/2)) / (x * (1+x/2)).
a(n) ~ sqrt(-sqrt(1 + 2*exp(-1)) + 1 + 2*exp(-1)) * n^(n-1) / (exp(n - 3/2) * (-1 + sqrt(1 + 2*exp(-1)))^n). (End)