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A356797 E.g.f. satisfies log(A(x)) = x * (exp(x) - 1) * A(x)^2.

%I A356797 #33 Feb 16 2025 08:34:03
%S A356797 1,0,2,3,64,305,6936,64897,1645008,24290289,692240680,14243244521,
%T A356797 456748635432,12105737521033,435619742434800,14112089558682585,
%U A356797 567134312211275296,21653262317886286817,966207399513747354072,42358800314758614030505
%N A356797 E.g.f. satisfies log(A(x)) = x * (exp(x) - 1) * A(x)^2.
%H A356797 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A356797 a(n) = n! * Sum_{k=0..floor(n/2)} (2*k+1)^(k-1) * Stirling2(n-k,k)/(n-k)!.
%F A356797 E.g.f.: A(x) = Sum_{k>=0} (2*k+1)^(k-1) * (x * (exp(x) - 1))^k / k!.
%F A356797 E.g.f.: A(x) = exp( -LambertW(2 * x * (1 - exp(x)))/2 ).
%F A356797 E.g.f.: A(x) = ( LambertW(2 * x * (1 - exp(x)))/(2 * x * (1 - exp(x))) )^(1/2).
%t A356797 m = 20; (* number of terms *)
%t A356797 CoefficientList[Exp[-(1/2)*LambertW[-2*(Exp[x]-1)*x]] + O[x]^m, x]*Range[0, m-1]! (* _Jean-François Alcover_, Sep 11 2022 *)
%o A356797 (PARI) a(n) = n!*sum(k=0, n\2, (2*k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!);
%o A356797 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (2*k+1)^(k-1)*(x*(exp(x)-1))^k/k!)))
%o A356797 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(2*x*(1-exp(x)))/2)))
%o A356797 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((lambertw(2*x*(1-exp(x)))/(2*x*(1-exp(x))))^(1/2)))
%Y A356797 Cf. A052506, A355843, A356798.
%Y A356797 Cf. A356788.
%K A356797 nonn
%O A356797 0,3
%A A356797 _Seiichi Manyama_, Aug 28 2022