A362775 - OEIS

%I #15 Nov 10 2023 07:48:13

%S 1,1,7,70,965,17216,379207,9969772,305154313,10668593008,419714689931,

%T 18358646058644,884070662867053,46486344447041032,2650567497877525423,

%U 162908800485532424236,10737607698626311094033,755571950776792829919968

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

%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)^2) ).

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

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

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

%F a(n) ~ 2^(n + 1/2) * sqrt(1 + 4*exp(-1) - sqrt(1 + 4*exp(-1))) * n^(n-1) / ((-1 + sqrt(1 + 4*exp(-1)))^(3/2) * (1 + 2*exp(-1) - sqrt(1 + 4*exp(-1)))^(n - 1/2) * exp(2*n-1)). (End)

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

%Y Cf. A082579, A362776.

%Y Cf. A052868, A361065.

%K nonn

%O 0,3

%A _Seiichi Manyama_, May 02 2023