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A356951 E.g.f. satisfies log(A(x)) = x^2/2 * (exp(x) - 1) * A(x).

%I A356951 #19 Feb 16 2025 08:34:04
%S A356951 1,0,0,3,6,10,285,1911,8848,155016,1931625,17006275,276807036,
%T A356951 4801114968,65672925409,1172625764415,24657199159440,460156401399376,
%U A356951 9560083801337793,230955040794126915,5393971086379904260,131545127670380245920,3587507216606547324321
%N A356951 E.g.f. satisfies log(A(x)) = x^2/2 * (exp(x) - 1) * A(x).
%H A356951 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A356951 a(n) = n! * Sum_{k=0..floor(n/3)} (k+1)^(k-1) * Stirling2(n-2*k,k)/(2^k * (n-2*k)!).
%F A356951 E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (x^2/2 * (exp(x) - 1))^k / k!.
%F A356951 E.g.f.: A(x) = exp( -LambertW(x^2/2 * (1 - exp(x))) ).
%F A356951 E.g.f.: A(x) = LambertW(x^2/2 * (1 - exp(x)))/(x^2/2 * (1 - exp(x))).
%t A356951 nmax = 22; A[_] = 1;
%t A356951 Do[A[x_] = Exp[x^2/2*(Exp[x] - 1)*A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
%t A356951 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 05 2024 *)
%o A356951 (PARI) a(n) = n!*sum(k=0, n\3, (k+1)^(k-1)*stirling(n-2*k, k, 2)/(2^k*(n-2*k)!));
%o A356951 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(x^2/2*(exp(x)-1))^k/k!)))
%o A356951 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(x^2/2*(1-exp(x))))))
%o A356951 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(x^2/2*(1-exp(x)))/(x^2/2*(1-exp(x)))))
%Y A356951 Cf. A052880, A355843, A356952.
%Y A356951 Cf. A000272, A354000, A356752, A356949.
%K A356951 nonn
%O A356951 0,4
%A A356951 _Seiichi Manyama_, Sep 06 2022