A370876 Expansion of e.g.f. (1/x) * Series_Reversion( x/(x + exp(x^3)) ).

1, 1, 2, 12, 120, 1320, 17640, 304920, 6249600, 143579520, 3711052800, 107762054400, 3455138332800, 120802387305600, 4583177081683200, 187766031131078400, 8256125218115174400, 387662886088250572800, 19364540503274942976000, 1025507260911983244595200

Offset: 0

Author

Seiichi Manyama, Mar 03 2024

Keywords

• nonn

Links

• Index entries for reversions of series

Crossrefs

Cf. A352410, A370875.

Cf. A360609.

Programs

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x/(x+exp(x^3)))/x))

PARI

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

Formula

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

E.g.f.: (LambertW( -3*x^3/(1-x)^3 ) / (-3*x^3))^(1/3).