A379878 - OEIS

%I #12 Jan 23 2025 04:27:40

%S 1,0,1,8,97,1544,30673,732752,20486401,656713520,23755416481,

%T 957430990328,42552022022497,2067669370359800,109058922249721585,

%U 6205740584180119424,378947624701223801089,24718152376534891564256,1715322065909959400535361,126186162087426817989206888

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

%F a(n) = -n! * Sum_{k=0..n} (-2*k-1)^(n-k-1) * binomial(3*k,k)/(n-k)!.

%F a(n) ~ (-1)^n * sqrt(-LambertW(-8/27) - 1) * 2^n * n^(n-1) / (3 * exp(n) * LambertW(-8/27)^(n + 1/2)). - _Vaclav Kotesovec_, Jan 23 2025

%t Table[-n! * Sum[(-2*k-1)^(n-k-1) * Binomial[3*k, k] / (n-k)!, {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Jan 23 2025 *)

%o (PARI) a(n) = -n!*sum(k=0, n, (-2*k-1)^(n-k-1)*binomial(3*k, k)/(n-k)!);

%Y Cf. A379871, A379876, A379877.

%Y Cf. A000166, A379879.

%K nonn,changed

%O 0,4

%A _Seiichi Manyama_, Jan 05 2025