A378669 G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x)^(4/3)/(1 - x*A(x)^(4/3)) )^3.

1, 3, 21, 187, 1878, 20277, 229806, 2696523, 32478204, 399230972, 4988220669, 63165060093, 808828667104, 10455471983550, 136255868388684, 1788233397919211, 23614059664575324, 313531617379965156, 4183068478829324388, 56052027108881747724, 754020313029799707018

Offset: 0

Author

Seiichi Manyama, Dec 02 2024

Keywords

• nonn

Crossrefs

Cf. A243667, A378610, A378668.

Cf. A378613.

Programs

PARI

a(n) = 3*sum(k=0, n, 2^k*(-1)^(n-k)*binomial(n, k)*binomial(4*n+k+3, n)/(4*n+k+3));

PARI

a(n, r=3, 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( 3/4 * Sum_{k>=1} A378613(k) * x^k/k ).

G.f.: B(x)^3 where B(x) is the g.f. of A243667.

a(n) = 3 * Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(n,k) * binomial(4*n+k+3,n)/(4*n+k+3).

a(n) = 3 * Sum_{k=0..n} binomial(4*n+k+3,k) * binomial(n-1,n-k)/(4*n+k+3).

G.f. A(x) satisfies A(x) = ( 1 + x*A(x)^(5/3)/(1 - x*A(x)^(4/3)) )^3.