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A356912 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^(x^2/2).

%I A356912 #19 Feb 16 2025 08:34:03
%S A356912 1,0,0,3,6,20,0,-126,-1260,18360,335160,4546080,26302320,-59501520,
%T A356912 -5703994296,-58549768200,371346066000,34962417322560,746101280831040,
%U A356912 8059680118183680,-93772611412099200,-5613314502242643840,-110940169654432087200
%N A356912 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^(x^2/2).
%H A356912 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A356912 a(n) = n! * Sum_{k=0..floor(n/3)} (-k+1)^(k-1) * |Stirling1(n-2*k,k)|/(2^k * (n-2*k)!).
%F A356912 E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (-x^2/2 * log(1-x))^k / k!.
%F A356912 E.g.f.: A(x) = exp( LambertW(-x^2/2 * log(1-x)) ).
%F A356912 E.g.f.: A(x) = -x^2/2 * log(1-x)/LambertW(-x^2/2 * log(1-x)).
%t A356912 nmax = 22; A[_] = 1;
%t A356912 Do[A[x_] = ((1 - x)^(-x^2/2))^(1/A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
%t A356912 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)
%o A356912 (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 A356912 (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 A356912 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-x^2/2*log(1-x)))))
%o A356912 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(-x^2/2*log(1-x)/lambertw(-x^2/2*log(1-x))))
%Y A356912 Cf. A351492, A356752.
%Y A356912 Cf. A356909, A356910.
%K A356912 sign
%O A356912 0,4
%A A356912 _Seiichi Manyama_, Sep 03 2022