login
E.g.f. satisfies A(x) = exp( x * (1+x/2) * A(x) ).
4

%I #18 Feb 16 2025 08:34:06

%S 1,1,4,25,230,2786,42112,764296,16209916,393678856,10777609556,

%T 328466815964,11031378197776,404830360798072,16118917055902312,

%U 692126238230304616,31882272572881781648,1568365865590875789824,82061348851406564851312

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

%H Seiichi Manyama, <a href="/A365053/b365053.txt">Table of n, a(n) for n = 0..379</a>

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

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

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

%F From _Vaclav Kotesovec_, Nov 10 2023: (Start)

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

%F a(n) ~ sqrt(-sqrt(1 + 2*exp(-1)) + 1 + 2*exp(-1)) * n^(n-1) / (exp(n - 3/2) * (-1 + sqrt(1 + 2*exp(-1)))^n). (End)

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

%Y Cf. A365054, A365055, A365056.

%Y Cf. A091485, A143740.

%K nonn,easy,changed

%O 0,3

%A _Seiichi Manyama_, Aug 19 2023