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A361068 E.g.f. satisfies A(x) = exp( x/((1-x) * A(x)^2) ).

%I A361068 #19 Feb 16 2025 08:34:04
%S A361068 1,1,-1,13,-127,2101,-41801,1030177,-29820127,995977801,-37660751569,
%T A361068 1590847310581,-74242656468575,3793664894534269,-210656932372422745,
%U A361068 12630986901470435401,-813335155262348743231,55977540398642247218449
%N A361068 E.g.f. satisfies A(x) = exp( x/((1-x) * A(x)^2) ).
%H A361068 Winston de Greef, <a href="/A361068/b361068.txt">Table of n, a(n) for n = 0..360</a>
%H A361068 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A361068 a(n) = n! * Sum_{k=0..n} (-2*k+1)^(k-1) * binomial(n-1,n-k)/k!.
%F A361068 E.g.f.: exp( LambertW(2*x/(1-x))/2 ).
%F A361068 E.g.f.: 1 / sqrt( (1-x)/(2*x) * LambertW(2*x/(1-x)) ).
%F A361068 a(n) ~ (-1)^(n+1) * 2^(-3/2) * exp(-1/2) * (2 - exp(-1))^(n + 1/2) * n^(n-1). - _Vaclav Kotesovec_, Apr 22 2024
%t A361068 nmax = 20; A[_] = 1;
%t A361068 Do[A[x_] = Exp[x/((1 - x)*A[x]^2)] + O[x]^(nmax+1) // Normal, {nmax}];
%t A361068 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)
%o A361068 (PARI) a(n) = n!*sum(k=0, n, (-2*k+1)^(k-1)*binomial(n-1, n-k)/k!);
%o A361068 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(2*x/(1-x))/2)))
%o A361068 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(1/sqrt((1-x)/(2*x)*lambertw(2*x/(1-x)))))
%Y A361068 Cf. A052868, A361065, A361066, A361067, A361069.
%K A361068 sign
%O A361068 0,4
%A A361068 _Seiichi Manyama_, Mar 01 2023