A391650 G.f. A(x) satisfies A(x) = 1/(1 - x/(1 - x*A(x)^4)^2).

1, 1, 3, 16, 97, 643, 4514, 32975, 248073, 1908922, 14954639, 118872632, 956360378, 7772627864, 63719954171, 526297294068, 4375423588669, 36584932651456, 307464725457800, 2595746654110011, 22004044556019121, 187216588916896334, 1598232286388255708

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Formula

G.f.: 1/(1 - x*B(x)^2), where B(x) is the g.f. of A391647.

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

Mathematica

Table[Sum[Binomial[4*n-3*k+1, k]*Binomial[n+k-1, n-k]/(4*n-3*k+1), {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Dec 20 2025 *)

Prog

(PARI) a(n, s=2, t=1, u=4) = sum(k=0, n, binomial(t*k+u*(n-k)+1, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+1));
(Magma) [&+[Binomial(4*n-3*k+1, k)*Binomial(n+k-1, n-k)/(4*n-3*k+1): k in [0..n]] : n in [0..30] ]; // Vincenzo Librandi, Dec 20 2025

Crossrefs

Cf. A002294, A364723, A391651.

Cf. A391647.

Sequence in context: A000271 A157016 A228792 * A295810 A233203 A074553

Adjacent sequences: A391647 A391648 A391649 * A391651 A391652 A391653

Keyword

nonn

Author

Seiichi Manyama, Dec 15 2025

Status

approved