A361213 E.g.f. satisfies A(x) = exp( 2*x*A(x) / (1+x) ).
1, 2, 8, 68, 848, 14192, 298048, 7546016, 223792640, 7612381952, 292216807424, 12497875215872, 589392367925248, 30386736933804032, 1700376343771136000, 102641314849948602368, 6648428846464054919168, 459977466799800897437696
Offset: 0
Keywords
Links
- Winston de Greef, Table of n, a(n) for n = 0..360
- Eric Weisstein's World of Mathematics, Lambert W-Function.
Programs
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PARI
a(n) = (-1)^n*n!*sum(k=0, n, (-2)^k*(k+1)^(k-1)*binomial(n-1, n-k)/k!);
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PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-2*x/(1+x))))) -
PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(-(1+x)/(2*x)*lambertw(-2*x/(1+x))))
Formula
a(n) = (-1)^n * n! * Sum_{k=0..n} (-2)^k * (k+1)^(k-1) * binomial(n-1,n-k)/k!.
E.g.f.: exp ( -LambertW(-2*x/(1+x)) ).
E.g.f.: -(1+x)/(2*x) * LambertW(-2*x/(1+x)).
a(n) ~ (2*exp(1) - 1)^(n + 1/2) * n^(n-1) / (sqrt(2) * exp(n - 1/2)). - Vaclav Kotesovec, Nov 10 2023