A382059 E.g.f. A(x) satisfies A(x) = exp(x*B(x*A(x))^3), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

1, 1, 7, 127, 3733, 152161, 7939261, 505087843, 37920697753, 3281899787137, 321700411900441, 35227497466867531, 4262151791317099285, 564639582580738851265, 81290104199287214904037, 12637400195063381931755731, 2109868901338065949399370161, 376504852688521502050554789889

Offset: 0

Author

Seiichi Manyama, Mar 13 2025

Keywords

• nonn

Crossrefs

Cf. A161629, A382058.

Cf. A364938, A382033.

Cf. A002293, A377548, A382034.

Programs

PARI

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

Formula

Let F(x) be the e.g.f. of A377548. F(x) = log(A(x))/x = B(x*A(x))^3.

E.g.f.: A(x) = exp( Series_Reversion( x*(1 - x*exp(x))^3 ) ).

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