A378668 G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x)^2/(1 - x*A(x)^2) )^2.

1, 2, 13, 112, 1104, 11778, 132374, 1543740, 18505996, 226632616, 2823110349, 35659080952, 455652487060, 5879489288828, 76502741016012, 1002670573618324, 13224761472453756, 175404372357915096, 2338003752387818372, 31302169754776944512, 420760252068869028028

Offset: 0

Author

Seiichi Manyama, Dec 02 2024

Keywords

• nonn

Crossrefs

Cf. A243667, A378610, A378669.

Cf. A010683, A211789, A378670.

Cf. A378613.

Programs

PARI

a(n) = 2*sum(k=0, n, 2^k*(-1)^(n-k)*binomial(n, k)*binomial(4*n+k+2, n)/(4*n+k+2));

PARI

a(n, r=2, s=1, t=5, u=4) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));

Formula

G.f.: exp( 1/2 * Sum_{k>=1} A378613(k) * x^k/k ).

G.f.: B(x)^2 where B(x) is the g.f. of A243667.

a(n) = 2 * Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(n,k) * binomial(4*n+k+2,n)/(4*n+k+2).

a(n) = 2 * Sum_{k=0..n} binomial(4*n+k+2,k) * binomial(n-1,n-k)/(4*n+k+2).

G.f. A(x) satisfies A(x) = ( 1 + x*A(x)^(5/2)/(1 - x*A(x)^2) )^2.