A370875 Expansion of e.g.f. (1/x) * Series_Reversion( x/(x + exp(x^2)) ).

1, 1, 4, 24, 228, 2820, 44400, 840000, 18669840, 475871760, 13698296640, 439402803840, 15545690233920, 601352177025600, 25251437978807040, 1143932660001331200, 55612090342967558400, 2887929114414030086400, 159548423949650274739200

Offset: 0

Author

Seiichi Manyama, Mar 03 2024

Keywords

• nonn

Links

• Index entries for reversions of series

Crossrefs

Cf. A352410, A370876.

Cf. A360601.

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x/(x+exp(x^2)))/x))

PARI

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

Formula

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

E.g.f.: sqrt(LambertW( -2*x^2/(1-x)^2 ) / (-2*x^2)).