A377595 E.g.f. satisfies A(x) = exp( x * A(x) / (1-x) ) / (1-x).

1, 2, 11, 103, 1377, 24101, 523813, 13636463, 414246017, 14396807161, 563682761541, 24559156435595, 1178780540094193, 61810491468265541, 3515914378433242997, 215647516162031069191, 14187967957218808201089, 996767406049512569338481, 74478502236949781909301253

Offset: 0

Author

Seiichi Manyama, Nov 14 2024

Keywords

• nonn

Links

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

Crossrefs

Cf. A362775, A377810.

Cf. A361598.

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x/(1-x)^2))/(1-x)))

PARI

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

Formula

E.g.f.: exp( -LambertW(-x/(1-x)^2) )/(1-x).

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

a(n) ~ sqrt(1 + 2*exp(-1) - sqrt(1 + 4*exp(-1))) * sqrt(1 + 4*exp(-1) - sqrt(1 + 4*exp(-1))) * 2^(n + 3/2) * n^(n-1) / ((sqrt(1 + 4*exp(-1)) - 1)^(5/2) * exp(n) * (2 + exp(1) - exp(1/2)*sqrt(4 + exp(1)))^n). - Vaclav Kotesovec, Aug 05 2025