A378828 G.f. A(x) satisfies A(x) = ( 1 + x*A(x)^(2/3)/(1 - x*A(x)^(4/3)) )^3.

1, 3, 12, 61, 354, 2220, 14649, 100218, 704373, 5055383, 36895221, 272975652, 2042782905, 15434838759, 117588475377, 902259691317, 6966487019220, 54086849181609, 421986564474946, 3306818224272945, 26015737668878523, 205405810986995869, 1627042895593132485

Offset: 0

Author

Seiichi Manyama, Dec 09 2024

Keywords

• nonn

Crossrefs

Cf. A349017, A378801, A378882.

Cf. A364739, A378889.

Programs

PARI

a(n, r=3, s=1, t=2, 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. A(x) satisfies A(x) = 1/( 1 - x*A(x)^(1/3)/(1 - x*A(x)^(4/3)) )^3.

G.f. A(x) satisfies A(x) = 1 + x * A(x)^(2/3) * (1 + A(x)^(1/3) + A(x)^(5/3)).

G.f.: A(x) = B(x)^3 where B(x) is the g.f. of A364739.

If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) / (1 - x*A(x)^(u/r))^s )^r, then a(n) = 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).