A360939 E.g.f. satisfies A(x) = exp( 2*x*A(x) / (1-x) ).
1, 2, 16, 212, 4016, 99952, 3096448, 115063328, 4993598464, 248071645952, 13888585800704, 865481914527232, 59426130052458496, 4458258196636276736, 362864617248019800064, 31848507841521274769408, 2998685833332127139299328, 301504120063370711801724928
Offset: 0
Keywords
Links
- Winston de Greef, Table of n, a(n) for n = 0..342
- Eric Weisstein's World of Mathematics, Lambert W-Function.
Programs
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PARI
a(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) = 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) ~ (1 + 2*exp(1))^(n + 1/2) * n^(n-1) / (sqrt(2) * exp(n - 1/2)). - Vaclav Kotesovec, Nov 10 2023