A378891 G.f. A(x) satisfies A(x) = ( 1 + x*A(x)^2/(1 + x*A(x)) )^3.

1, 3, 18, 142, 1278, 12429, 127223, 1350456, 14729628, 164079982, 1858781652, 21348787587, 248021665720, 2909439099543, 34413536180688, 409984974779725, 4915119769384221, 59252402698999209, 717819918438472134, 8734481867945979183, 106703642464149880248

Offset: 0

Author

Seiichi Manyama, Dec 10 2024

Keywords

• nonn

Crossrefs

Cf. A371542, A378889, A378890.

Cf. A378892.

Programs

PARI

a(n, r=3, s=-1, t=6, u=3) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r));

Formula

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

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

G.f.: A(x) = B(x)^3 where B(x) is the g.f. of A378892.

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(s*k,n-k)/(t*k+u*(n-k)+r).