A372158 E.g.f. A(x) satisfies A(x) = exp( 2 * x * A(x)^(1/2) / (1 - x) ).
1, 2, 12, 110, 1368, 21602, 415036, 9416094, 246730448, 7340456258, 244615296564, 9030708939518, 365998814372824, 16159576541122146, 772216069907880812, 39715949460883093598, 2187682975276318552224, 128508919233259720967810
Offset: 0
Keywords
Links
- Eric Weisstein's World of Mathematics, Lambert W-Function.
Programs
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PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-2*lambertw(-x/(1-x))))) -
PARI
a(n, r=2, s=1, t=1, u=0) = r*n!*sum(k=0, n, (t*k+u*(n-k)+r)^(k-1)*binomial(n+(s-1)*k-1, n-k)/k!);
Formula
E.g.f.: A(x) = exp( -2 * LambertW(-x / (1-x)) ).
If e.g.f. satisfies A(x) = exp( r*x*A(x)^(t/r) / (1 - x*A(x)^(u/r))^s ), then a(n) = r * n! * Sum_{k=0..n} (t*k+u*(n-k)+r)^(k-1) * binomial(n+(s-1)*k-1,n-k)/k!.
a(n) ~ 2 * exp(2) * (1 + exp(-1))^(n + 1/2) * n^(n-1). - Vaclav Kotesovec, Aug 05 2025