A371813 a(n) = Sum_{k=0..n} (-1)^k * binomial(3*n-k-1,n-k).

1, 1, 7, 40, 239, 1461, 9076, 57044, 361711, 2309467, 14827487, 95630272, 619111172, 4021011580, 26187682024, 170960159100, 1118406332655, 7330011083079, 48119501497909, 316354663355384, 2082573599282359, 13726029056757029, 90565080767425744

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

a(n) = [x^n] 1/((1+x) * (1-x)^(2*n)).

a(n) = binomial(3*n-1, n)*hypergeom([1, -n], [1-3*n], -1). - Stefano Spezia, Apr 07 2024

From Vaclav Kotesovec, Apr 07 2024: (Start)

Recurrence: 8*n*(2*n - 1)*(28*n^2 - 87*n + 67)*a(n) = 2*(1456*n^4 - 6008*n^3 + 8593*n^2 - 4949*n + 960)*a(n-1) + 3*(3*n - 5)*(3*n - 4)*(28*n^2 - 31*n + 8)*a(n-2).

a(n) ~ 3^(3*n + 1/2) / (sqrt(Pi*n) * 2^(2*n+2)). (End)

a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(3*n,k). - Seiichi Manyama, Jul 30 2025

G.f.: g/((-1+2*g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764. - Seiichi Manyama, Aug 13 2025

a(n) = Sum_{k=0..n} (-1)^k * 2^(n-k) * binomial(3*n,k) * binomial(3*n-k-1,n-k). - Seiichi Manyama, Aug 15 2025

G.f.: 1/(1 - x*g^2*(-3+4*g)) where g = 1+x*g^3 is the g.f. of A001764. - Seiichi Manyama, Aug 17 2025

a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-2*k-2,n-2*k). - Seiichi Manyama, Nov 12 2025

Prog

(PARI) a(n) = sum(k=0, n, (-1)^k*binomial(3*n-k-1, n-k));

Crossrefs

Cf. A072547, A371814.

Cf. A183160, A390543.

Cf. A005809, A066380, A165817, A385004, A386700.

Sequence in context: A099459 A243875 A360101 * A051814 A154968 A121582

Adjacent sequences: A371810 A371811 A371812 * A371814 A371815 A371816

Keyword

nonn

Author

Seiichi Manyama, Apr 06 2024

Status

approved