A377811 - OEIS

%I #9 Nov 08 2024 11:37:30

%S 1,4,27,283,4217,82971,2041855,60475885,2096566449,83324680435,

%T 3736041351311,186594364199277,10274269171279657,618386703880855339,

%U 40393224245061185919,2846030947359659421901,215160957844217080056161,17373449685399138641312739,1492298627191467511376377999

%N E.g.f. satisfies A(x) = exp(x * A(x)) / (1 - x)^3.

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

%F E.g.f.: exp( -LambertW(-x/(1-x)^3) )/(1-x)^3.

%F E.g.f.: -LambertW(-x/(1-x)^3)/x.

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

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x/(1-x)^3))/(1-x)^3))

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

%Y Cf. A352410, A377810.

%Y Cf. A082030, A367789, A377743.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Nov 08 2024