A362836 Expansion of e.g.f. 1/(1 + LambertW(-x * (exp(x) - 1))).
1, 0, 2, 3, 52, 245, 4086, 36547, 663832, 8984313, 184262770, 3334315391, 77900601780, 1751855308645, 46508427942718, 1241853335819475, 37195023972070576, 1144511291020453361, 38337497638919397738, 1331709923436162817447
Offset: 0
Keywords
Programs
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PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(-x*(exp(x)-1)))))
Formula
a(n) = n! * Sum_{k=0..floor(n/2)} k^k * Stirling2(n-k,k)/(n-k)!.
a(n) ~ n^n / (sqrt(exp(r)*(1+r) - 1) * r^(n + 1/2) * exp(n + 1/2)), where r = 0.528399250336668412340528181936966763473... is the root of the equation exp(1+r)-exp(1) = 1/r. - Vaclav Kotesovec, May 19 2025