A380042 E.g.f. A(x) satisfies A(x) = 1/sqrt( 1 - 2*x*exp(x*A(x)^2) ).

1, 1, 5, 48, 697, 13640, 336771, 10053778, 352334753, 14183529480, 645073504435, 32715111226886, 1830671281889649, 112049330303532388, 7446824171300128811, 534068807341887943770, 41111698162393482004801, 3381089519620006418116976, 295869084136630532211207843

Offset: 0

Author

Seiichi Manyama, Jan 10 2025

Keywords

• nonn

Crossrefs

Cf. A380015, A380035.

Cf. A161633, A380043.

Cf. A201470.

Programs

PARI

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

PARI

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

Formula

E.g.f.: sqrt( (1/x) * Series_Reversion(x/(1 + 2*x*exp(x))) ).

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