A380095 E.g.f. A(x) satisfies A(x) = 1/sqrt( 1 - 2*x*A(x)^2*exp(x*A(x)^2) ).

1, 1, 9, 156, 4129, 147880, 6696591, 367141306, 23648581713, 1750754472840, 146492770433095, 13672570280741086, 1408330043282040825, 158697952371711709060, 19420527592823261136519, 2564857285665551372127570, 363619232307437704055993761, 55079007956127598819416831088

Offset: 0

Author

Seiichi Manyama, Jan 12 2025

Keywords

• nonn

Crossrefs

Cf. A213644, A380096.

Cf. A380042, A379688.

Programs

PARI

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

Formula

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

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

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