A380077 Expansion of e.g.f. (1/x) * Series_Reversion( x * sqrt(1 - 2*x*exp(x)) ).

1, 1, 7, 87, 1621, 40485, 1271841, 48220207, 2143450009, 109350344745, 6298638659245, 404371344546411, 28633701543626037, 2217105596852342989, 186362307297569836993, 16901012222196104542695, 1644911203243501609414321, 171017059743998995011125457, 18916512667390427993433246357

Offset: 0

Author

Seiichi Manyama, Jan 11 2025

Keywords

• nonn

Crossrefs

Cf. A213644, A380078.

Cf. A380035.

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x*sqrt(1-2*x*exp(x)))/x))

PARI

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

Formula

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

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

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