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%I #25 Mar 04 2024 08:47:35
%S 1,1,2,10,70,646,7576,106744,1761628,33361948,712950616,16976294776,
%T 445751093800,12795850109992,398697898011232,13401365473319776,
%U 483376669737381136,18623161719254837008,763300232417720682784,33163224556779213475744
%N E.g.f. satisfies A(x) = exp(x + x^2/2 * A(x)^2).
%H Seiichi Manyama, <a href="/A362474/b362474.txt">Table of n, a(n) for n = 0..392</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F E.g.f.: exp(x - LambertW(-x^2 * exp(2*x))/2) = sqrt(-LambertW(-x^2*exp(2*x))/x^2).
%F a(n) = n! * Sum_{k=0..floor(n/2)} (1/2)^k * (2*k+1)^(n-k-1) / (k! * (n-2*k)!).
%F a(n) ~ sqrt(1 + LambertW(exp(-1/2))) * n^(n-1) / (sqrt(2) * exp(n) * LambertW(exp(-1/2))^(n+1)). - _Vaclav Kotesovec_, Nov 10 2023
%t nmax = 20; A[_] = 1;
%t Do[A[x_] = Exp[x + x^2/2*A[x]^2] + O[x]^(nmax+1) // Normal, {nmax}];
%t CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-x^2*exp(2*x))/2)))
%Y Column k=1 of A362483.
%Y Cf. A362478, A362491.
%Y Cf. A143768, A362475.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Apr 21 2023
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