A360939 - OEIS

%I #32 Nov 10 2023 07:09:30

%S 1,2,16,212,4016,99952,3096448,115063328,4993598464,248071645952,

%T 13888585800704,865481914527232,59426130052458496,4458258196636276736,

%U 362864617248019800064,31848507841521274769408,2998685833332127139299328,301504120063370711801724928

%N E.g.f. satisfies A(x) = exp( 2*x*A(x) / (1-x) ).

%H Winston de Greef, <a href="/A360939/b360939.txt">Table of n, a(n) for n = 0..342</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.

%F a(n) = n! * Sum_{k=0..n} 2^k * (k+1)^(k-1) * binomial(n-1,n-k)/k!.

%F E.g.f.: exp ( -LambertW(-2*x/(1-x)) ).

%F E.g.f.: -(1-x)/(2*x) * LambertW(-2*x/(1-x)).

%F a(n) ~ (1 + 2*exp(1))^(n + 1/2) * n^(n-1) / (sqrt(2) * exp(n - 1/2)). - _Vaclav Kotesovec_, Nov 10 2023

%o (PARI) a(n) = n!*sum(k=0, n, 2^k*(k+1)^(k-1)*binomial(n-1, n-k)/k!);

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-2*x/(1-x)))))

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(-(1-x)/(2*x)*lambertw(-2*x/(1-x))))

%Y Cf. A052868, A361212.

%Y Cf. A361065.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Mar 04 2023