A378090 E.g.f. satisfies A(x) = exp(x * (1-x)^2 * A(x)) / (1-x)^3.
1, 4, 23, 181, 1889, 25411, 427615, 8736337, 210911489, 5882285971, 186121646831, 6585885144697, 257640988064641, 11039620794801691, 514147575711741119, 25858553659455655201, 1396703647943164718081, 80633376290492591578147, 4954794080385073122030799
Offset: 0
Keywords
Links
- Eric Weisstein's World of Mathematics, Lambert W-Function.
Programs
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Mathematica
terms = 19; A[] = 0; Do[A[x] = Exp[x*(1-x)^2*A[x]]/(1-x)^3 + O[x]^terms // Normal, terms]; CoefficientList[A[x], x]Range[0,terms-1]! (* Stefano Spezia, Mar 24 2025 *)
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PARI
a(n) = n!*sum(k=0, n, (k+1)^(k-1)*binomial(n+2, n-k)/k!);
Formula
E.g.f.: exp( -LambertW(-x/(1-x)) )/(1-x)^3.
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(n+2,n-k)/k!.
a(n) ~ n^(n-1) * (1 + exp(1))^(n + 7/2) / exp(n + 5/2). - Vaclav Kotesovec, Aug 05 2025