A376575 E.g.f. A(x) satisfies A(x) = exp(x*A(x)/(1 - x^2)).
1, 1, 3, 22, 197, 2376, 35047, 619984, 12772041, 300946816, 7985754251, 235775556864, 7668016756237, 272432946304000, 10499615465565423, 436328344923744256, 19450112299718461073, 925826421005833568256, 46870797202270907609107, 2514801570124507348271104
Offset: 0
Keywords
Links
- Eric Weisstein's World of Mathematics, Lambert W-Function.
Programs
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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)!);
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PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x/(1-x^2)))))
Formula
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)!.
E.g.f.: exp( -LambertW(-x/(1-x^2)) ).
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