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

Offset

0,2

Links

Table of n, a(n) for n=0..18.

Eric Weisstein's World of Mathematics, Lambert W-Function.

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

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 *)

Prog

(PARI) a(n) = n!*sum(k=0, n, (k+1)^(k-1)*binomial(n+3, n-k)/k!);

Crossrefs

Cf. A323772, A352410, A378090.

Sequence in context: A325153 A129890 A316158 * A120733 A218496 A144792

Adjacent sequences: A378088 A378089 A378090 * A378092 A378093 A378094

Keyword

nonn

Author

Seiichi Manyama, Nov 16 2024

Status

approved