A362771 E.g.f. satisfies A(x) = exp( x * (1+x) * A(x) ).

1, 1, 5, 34, 353, 4756, 80107, 1617358, 38145473, 1029745576, 31326858611, 1060716408874, 39571357618465, 1612919873514028, 71321521181852411, 3400790769764598886, 173950205958460627073, 9501239617356541012432, 551961456374529522954595

Offset: 0

Author

Seiichi Manyama, May 02 2023

Keywords

• nonn

Links

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

Crossrefs

Cf. A052868, A125500.

Programs

Mathematica

nmax = 20; A[_] = 1;
Do[A[x_] = Exp[x*(1 + x)*A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

PARI

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

Formula

E.g.f.: exp( -LambertW(-x * (1+x)) ).

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

a(n) ~ sqrt(2 + 8*exp(-1) - 2*sqrt(1 + 4*exp(-1))) * 2^(n-1) * n^(n-1) / ((sqrt(1 + 4*exp(-1)) - 1)^n * exp(n - 3/2)). - Vaclav Kotesovec, May 03 2023