A379861 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(x)/(1 + x*exp(x))^2 ).

1, 1, 5, 38, 441, 6714, 128245, 2943562, 79049201, 2432351618, 84408126621, 3261942050058, 138946757581225, 6468600047278498, 326782092756236741, 17805164917279808234, 1040857709162817298401, 64983981546315031200258, 4315627103007355018430509

Offset

0,3

Links

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

Index entries for reversions of series

Formula

E.g.f. A(x) satisfies A(x) = exp(-x*A(x)) * (1 + x * A(x) * exp(x*A(x)))^2.

a(n) = (n!/(n+1)) * Sum_{k=0..n} (-n+k-1)^(n-k) * binomial(2*n+2,k)/(n-k)!.

a(n) ~ 2^(2*n + 5/2) * n^(n-1) / ((1 + 1/LambertW(1)) * exp(n)). - Vaclav Kotesovec, Feb 02 2026

Mathematica

Table[n!/(n+1) * Sum[(-n+k-1)^(n-k) * Binomial[2*n+2, k]/(n-k)!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Feb 02 2026 *)

Prog

(PARI) a(n) = n!*sum(k=0, n, (-n+k-1)^(n-k)*binomial(2*n+2, k)/(n-k)!)/(n+1);

Crossrefs

Cf. A377553, A379862.

Sequence in context: A216858 A338867 A110467 * A221845 A299054 A095230

Adjacent sequences: A379858 A379859 A379860 * A379862 A379863 A379864

Keyword

nonn

Author

Seiichi Manyama, Jan 04 2025

Status

approved