A378091 E.g.f. satisfies A(x) = exp(x * (1-x)^3 * A(x)) / (1-x)^4.
1, 5, 33, 280, 3009, 40456, 670351, 13428794, 318341841, 8747362540, 273595272231, 9595433139238, 372786185735497, 15885841209363152, 736549352642825247, 36906793949098033906, 1987212351128733260577, 114415986259681057007956, 7014281833059332148174007
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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)^3*A[x]]/(1-x)^4 + 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+3, n-k)/k!);
Formula
E.g.f.: exp( -LambertW(-x/(1-x)) )/(1-x)^4.
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(n+3,n-k)/k!.
a(n) ~ n^(n-1) * (1 + exp(1))^(n + 9/2) / exp(n + 7/2). - Vaclav Kotesovec, Aug 05 2025