A364747 G.f. A(x) satisfies A(x) = 1 + x*A(x)^4 / (1 - x*A(x)).

1, 1, 5, 32, 234, 1854, 15490, 134380, 1198944, 10931761, 101412677, 954155059, 9083120975, 87326765375, 846709605539, 8269910074087, 81291388929027, 803592049667495, 7983612883739843, 79671910265120574, 798283229227457304, 8027625597750959053

Offset: 0

Author

Seiichi Manyama, Aug 05 2023

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..972

Crossrefs

Cf. A000108, A001003, A108447, A364748, A378691, A378692.

Cf. A002294, A243659, A364765, A364792.

Programs

PARI

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

PARI

a(n, r=1, s=1, t=4, u=1) = 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)); \\ Seiichi Manyama, Dec 05 2024

Formula

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

From Seiichi Manyama, Dec 05 2024: (Start)

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

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). (End)