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A335945 E.g.f. A(x) satisfies A(x) = exp(x*A(x)/(1 + x)).

%I A335945 #17 Feb 16 2025 08:34:00
%S A335945 1,1,1,4,17,116,907,9010,102097,1348408,19939571,330204854,6015657529,
%T A335945 120016789348,2597201945899,60667591974826,1520434054966433,
%U A335945 40710815980598000,1159627208850209251,35018022339726428926,1117395892399939407241,37569709612314269554396
%N A335945 E.g.f. A(x) satisfies A(x) = exp(x*A(x)/(1 + x)).
%H A335945 Seiichi Manyama, <a href="/A335945/b335945.txt">Table of n, a(n) for n = 0..414</a>
%H A335945 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A335945 E.g.f.: -(1 + x) * LambertW(-x/(1 + x)) / x.
%F A335945 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n-1,k-1) * (k+1)^(k-1) * n! / k!.
%F A335945 a(n) ~ (exp(1) - 1)^(n + 1/2) * n^(n-1) / exp(n - 1/2). - _Vaclav Kotesovec_, Jul 01 2020
%F A335945 E.g.f.: exp ( -LambertW(-x/(1+x)) ). - _Seiichi Manyama_, Mar 05 2023
%t A335945 nmax = 21; A[_] = 0; Do[A[x_] = Exp[x A[x]/(1 + x)] + O[x]^(nmax + 1) // Normal, nmax + 1];CoefficientList[A[x], x] Range[0, nmax]!
%t A335945 nmax = 21; CoefficientList[Series[-(1 + x) LambertW[-x/(1 + x)]/x, {x, 0, nmax}], x] Range[0, nmax]!
%t A335945 Table[Sum[(-1)^(n - k) Binomial[n - 1, k - 1] (k + 1)^(k - 1) n!/k!, {k, 0, n}], {n, 0, 21}]
%o A335945 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x/(1+x))))) \\ _Seiichi Manyama_, Mar 05 2023
%Y A335945 Cf. A052868, A060356, A108919.
%K A335945 nonn
%O A335945 0,4
%A A335945 _Ilya Gutkovskiy_, Jul 01 2020