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A361065 E.g.f. satisfies A(x) = exp( (x/(1-x)) * A(x)^2 ).

%I A361065 #29 Feb 16 2025 08:34:04
%S A361065 1,1,7,85,1521,36421,1097743,39968601,1707558401,83777885929,
%T A361065 4643185678551,286930307457949,19562851003118833,1458832806486727725,
%U A361065 118121195050068075167,10320576944751955718881,967863775658734350214017,96970880819175875321264209
%N A361065 E.g.f. satisfies A(x) = exp( (x/(1-x)) * A(x)^2 ).
%H A361065 Winston de Greef, <a href="/A361065/b361065.txt">Table of n, a(n) for n = 0..342</a>
%H A361065 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A361065 a(n) = n! * Sum_{k=0..n} (2*k+1)^(k-1) * binomial(n-1,n-k)/k!.
%F A361065 E.g.f.: exp( -LambertW(-2*x/(1-x))/2 ).
%F A361065 E.g.f.: sqrt( -(1-x)/(2*x) * LambertW(-2*x/(1-x)) ).
%F A361065 a(n) ~ (1 + 2*exp(1))^(n + 1/2) * n^(n-1) / (2^(3/2) * exp(n)). - _Vaclav Kotesovec_, Mar 02 2023
%p A361065 A361065 := proc(n)
%p A361065     add((2*k+1)^(k-1)*binomial(n-1,n-k)/k!,k=0..n) ;
%p A361065     %*n! ;
%p A361065 end proc:
%p A361065 seq(A361065(n),n=0..10) ; # _R. J. Mathar_, Mar 02 2023
%t A361065 nmax = 20; A[_] = 1;
%t A361065 Do[A[x_] = Exp[(x/(1 - x))*A[x]^2] + O[x]^(nmax+1) // Normal, {nmax}];
%t A361065 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)
%o A361065 (PARI) a(n) = n!*sum(k=0, n, (2*k+1)^(k-1)*binomial(n-1, n-k)/k!);
%o A361065 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-2*x/(1-x))/2)))
%o A361065 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sqrt(-(1-x)/(2*x)*lambertw(-2*x/(1-x)))))
%Y A361065 Cf. A052868, A361066, A361067, A361068, A361069.
%K A361065 nonn
%O A361065 0,3
%A A361065 _Seiichi Manyama_, Mar 01 2023