A377860 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x)^2 * exp(x) ).

1, 1, 5, 44, 577, 10104, 222133, 5886880, 182775969, 6509571200, 261665344261, 11720054882304, 578878362625825, 31259890045425664, 1832295378792935925, 115862322601669627904, 7861907382202262095297, 569837358810005613281280, 43939338917141224534941829

Offset

0,3

Links

Table of n, a(n) for n=0..18.

Index entries for reversions of series

Formula

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

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

a(n) ~ (sqrt(2)-1) * n^(n-1) / (2^(n + 3/2) * exp(sqrt(2)*n + sqrt(2) - 1) * (5*sqrt(2)-7)^(n+1)). - Vaclav Kotesovec, Jan 31 2026

Mathematica

Table[n! * Sum[(-1)^k * (n+1)^(k-1) * Binomial[3*n-k+1, n-k]/k!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 31 2026 *)

Prog

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

Crossrefs

Cf. A377859, A377861.

Cf. A377832.

Sequence in context: A106273 A349836 A052803 * A201923 A222059 A336290

Adjacent sequences: A377857 A377858 A377859 * A377861 A377862 A377863

Keyword

nonn

Author

Seiichi Manyama, Nov 09 2024

Status

approved