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A357009 E.g.f. satisfies log(A(x)) = (exp(x) - 1)^2 * A(x).

%I A357009 #34 Feb 16 2025 08:34:04
%S A357009 1,0,2,6,50,390,4322,53046,782210,12920550,241747682,5000171286,
%T A357009 113961184130,2830240421190,76196913418082,2209152734071926,
%U A357009 68655746019566210,2276606079902438310,80244521295497399522,2995966456305973559766,118119901491333724203650
%N A357009 E.g.f. satisfies log(A(x)) = (exp(x) - 1)^2 * A(x).
%H A357009 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A357009 a(n) = Sum_{k=0..floor(n/2)} (2*k)! * (k+1)^(k-1) * Stirling2(n,2*k)/k!.
%F A357009 E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (exp(x) - 1)^(2*k) / k!.
%F A357009 E.g.f.: A(x) = exp( -LambertW(-(exp(x) - 1)^2) ).
%F A357009 E.g.f.: A(x) = -LambertW(-(exp(x) - 1)^2)/(exp(x) - 1)^2.
%F A357009 a(n) ~ sqrt(1 + exp(1/2)) * 2^n * n^(n-1) / (exp(n-1) * (2*log(1 + exp(1/2)) - 1)^(n - 1/2)). - _Vaclav Kotesovec_, Sep 27 2023
%t A357009 nmax = 20; A[_] = 1;
%t A357009 Do[A[x_] = Exp[(-1 + Exp[x])^2*A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
%t A357009 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)
%o A357009 (PARI) a(n) = sum(k=0, n\2, (2*k)!*(k+1)^(k-1)*stirling(n, 2*k, 2)/k!);
%o A357009 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(exp(x)-1)^(2*k)/k!)))
%o A357009 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-(exp(x)-1)^2))))
%o A357009 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(-lambertw(-(exp(x)-1)^2)/(exp(x)-1)^2))
%Y A357009 Cf. A052880, A357010.
%Y A357009 Cf. A052859, A357024.
%K A357009 nonn
%O A357009 0,3
%A A357009 _Seiichi Manyama_, Sep 09 2022