A371038 - OEIS

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

%S 1,1,4,18,132,1140,12720,164640,2514960,43500240,850076640,

%T 18418609440,439831909440,11457415569600,323707663319040,

%U 9854548934630400,321709145793235200,11209975693710393600,415330670608805952000,16303720885477254028800

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

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

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

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

%F a(n) ~ exp(2) * sqrt(1 + 4*exp(1) - sqrt(1 + 4*exp(1))) * 2^(n + 3/2) * n^(n-1) / ((1 + 2*exp(1) - sqrt(1 + 4*exp(1)))*(-1 + sqrt(1 + 4*exp(1)))^(n+1)). - _Vaclav Kotesovec_, Mar 12 2024

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(-x^2/(1-x))/(-x^2)))

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

%Y Cf. A352410, A371039.

%K nonn,changed

%O 0,3

%A _Seiichi Manyama_, Mar 09 2024