A371021 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x^3/6*exp(x)) ).

1, 0, 0, 1, 4, 10, 80, 1015, 9016, 80724, 1092120, 16872405, 246966940, 3932454526, 73869476044, 1485097614455, 30688224287280, 682450482838440, 16508839426673136, 420562937260614249, 11193327347979937140, 315276822746559147250, 9383980947735649740100

Offset: 0

Author

Seiichi Manyama, Mar 08 2024

Keywords

• nonn

Links

• Index entries for reversions of series

Crossrefs

Cf. A370926, A371019.

Programs

Mathematica

Join[{1}, Table[(n!/(n + 1))*Sum[k^(n - 3*k)*Binomial[n + 1, k]/(6^k*(n - 3*k)!), {k, 0, Floor[n/3]}], {n, 30}]] (* Wesley Ivan Hurt, Aug 05 2025 *)

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x/(1+x^3/6*exp(x)))/x))

PARI

a(n) = n!*sum(k=0, n\3, k^(n-3*k)*binomial(n+1, k)/(6^k*(n-3*k)!))/(n+1);

Formula

a(n) = (n!/(n+1)) * Sum_{k=0..floor(n/3)} k^(n-3*k) * binomial(n+1,k)/(6^k * (n-3*k)!).