A376438 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x^2*(exp(x) - 1))^2 ).

1, 0, 0, 12, 24, 40, 10860, 85764, 446992, 57788784, 1008736020, 10835748220, 965748698904, 28637803537512, 519426455756572, 37968161216666100, 1626852405783259680, 44177643556314690784, 2957776991432290423332, 163869985958022692795628, 6132727345895339422510120, 405409522521171206216078040

Offset: 0

Author

Seiichi Manyama, Sep 22 2024

Keywords

• nonn

Links

• Index entries for reversions of series

Crossrefs

Cf. A376381, A376389.

Cf. A375662.

Programs

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-x^2*(exp(x)-1))^2)/x))

PARI

a(n) = 2*n!*sum(k=0, n\3, (2*n+k+1)!*stirling(n-2*k, k, 2)/(n-2*k)!)/(2*n+2)!;

Formula

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

a(n) = (2 * n!/(2*n+2)!) * Sum_{k=0..floor(n/3)} (2*n+k+1)! * Stirling2(n-2*k,k)/(n-2*k)!.