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E.g.f. A(x) satisfies A(x) = exp(x*A(x)/(1 - x^2)).
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%I #14 Sep 29 2024 08:54:39

%S 1,1,3,22,197,2376,35047,619984,12772041,300946816,7985754251,

%T 235775556864,7668016756237,272432946304000,10499615465565423,

%U 436328344923744256,19450112299718461073,925826421005833568256,46870797202270907609107,2514801570124507348271104

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.

%F a(n) = n! * Sum_{k=0..floor(n/2)} (n-2*k+1)^(n-2*k-1) * binomial(n-k-1,k)/(n-2*k)!.

%F E.g.f.: exp( -LambertW(-x/(1-x^2)) ).

%F a(n) ~ (1 + 4*exp(-2))^(1/4) * 2^n * n^(n-1) / ((sqrt(1 + 4*exp(-2)) - 1)^n * exp(2*n-1)). - _Vaclav Kotesovec_, Sep 29 2024

%o (PARI) a(n) = n!*sum(k=0, n\2, (n-2*k+1)^(n-2*k-1)*binomial(n-k-1, k)/(n-2*k)!);

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x/(1-x^2)))))

%Y Cf. A052868, A376576.

%Y Cf. A088009, A376327.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Sep 28 2024