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A356752 E.g.f. satisfies A(x) = 1/(1 - x)^(x^2/2 * A(x)).

%I A356752 #27 Feb 16 2025 08:34:03
%S A356752 1,0,0,3,6,20,360,2394,17220,260280,3076920,35980560,595686960,
%T A356752 9760411440,159321570408,3093987619800,63314740616400,
%U A356752 1318245318411840,30240056863978560,736919729169603840,18522487833889334400,495842871278901363840,14014346231616983128800
%N A356752 E.g.f. satisfies A(x) = 1/(1 - x)^(x^2/2 * A(x)).
%H A356752 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A356752 a(n) = n! * Sum_{k=0..floor(n/3)} (k+1)^(k-1) * |Stirling1(n-2*k,k)|/(2^k * (n-2*k)!).
%F A356752 E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (-x^2/2 * log(1-x))^k / k!.
%F A356752 E.g.f.: A(x) = exp( -LambertW(x^2/2 * log(1-x)) ).
%F A356752 E.g.f.: A(x) = LambertW(x^2/2 * log(1-x))/(x^2/2 * log(1-x)).
%t A356752 nmax = 22; A[_] = 1;
%t A356752 Do[A[x_] = 1/(1 - x)^(x^2/2*A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
%t A356752 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)
%o A356752 (PARI) a(n) = n!*sum(k=0, n\3, (k+1)^(k-1)*abs(stirling(n-2*k, k, 1))/(2^k*(n-2*k)!));
%o A356752 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(-x^2/2*log(1-x))^k/k!)))
%o A356752 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(x^2/2*log(1-x)))))
%o A356752 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(x^2/2*log(1-x))/(x^2/2*log(1-x))))
%Y A356752 Cf. A351492, A356912.
%Y A356752 Cf. A355842, A356753.
%K A356752 nonn
%O A356752 0,4
%A A356752 _Seiichi Manyama_, Sep 03 2022