A376389 Expansion of e.g.f. (1/x) * Series_Reversion( x*(2 - exp(x))^2 ).

1, 2, 16, 236, 5172, 151452, 5568452, 246816236, 12817081828, 763506280700, 51333645252228, 3845783934171852, 317719919221661540, 28697779828343464412, 2813593953407672094724, 297587218343306095847084, 33775895041558685181041892, 4094844200848292606224524732

Offset: 0

Author

Seiichi Manyama, Sep 22 2024

Keywords

• nonn

Links

• Index entries for reversions of series

Crossrefs

Cf. A005649, A367134.

Programs

PARI

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

PARI

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

Formula

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

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A367134.

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

a(n) ~ 2^n * LambertW(exp(1/2))^(2*n + 2)*n^(n-1) / (sqrt(1 + LambertW(exp(1/2))) * exp(n) * (2*LambertW(exp(1/2)) - 1)^(3*n + 2)). - Vaclav Kotesovec, Sep 22 2024