%I #17 Mar 04 2024 07:47:07

%S 1,1,5,34,353,4756,80107,1617358,38145473,1029745576,31326858611,

%T 1060716408874,39571357618465,1612919873514028,71321521181852411,

%U 3400790769764598886,173950205958460627073,9501239617356541012432,551961456374529522954595

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.

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

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

%F 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

%t nmax = 20; A[_] = 1;

%t Do[A[x_] = Exp[x*(1 + x)*A[x]] + O[x]^(nmax+1) // Normal, {nmax}];

%t CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)

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

%Y Cf. A052868, A125500.

%K nonn

%O 0,3

%A _Seiichi Manyama_, May 02 2023