A385004 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n,k).

1, 5, 31, 200, 1311, 8665, 57556, 383556, 2561871, 17140007, 114819351, 769925568, 5166845124, 34696155564, 233113911208, 1566926561740, 10536427052463, 70872688450083, 476854924775869, 3209222876463192, 21602639249766951, 145444151677134153, 979397744169608784

Offset: 0

Author

Seiichi Manyama, Jul 30 2025

Keywords

• nonn
• easy

Crossrefs

Cf. A000244, A100192, A385498, A386699.

Cf. A005809, A066380, A165817, A371813, A386700.

Cf. A001764, A386617.

Programs

Mathematica

Table[(27/4)^n - Binomial[3*n, n] * (-1 + Hypergeometric2F1[1, -2*n, 1 + n, -1/2]), {n, 0, 25}] (* Vaclav Kotesovec, Jul 30 2025 *)

PARI

a(n) = sum(k=0, n, 2^(n-k)*binomial(3*n, k));

Formula

a(n) = [x^n] 1/((1-3*x) * (1-x)^(2*n)).

a(n) = Sum_{k=0..n} 3^(n-k) * binomial(2*n+k-1,k).

From Vaclav Kotesovec, Jul 30 2025: (Start)

Recurrence: 8*n*(2*n - 1)*(15*n - 23)*a(n) = 6*(540*n^3 - 1503*n^2 + 1239*n - 320)*a(n-1) - 81*(3*n - 5)*(3*n - 4)*(15*n - 8)*a(n-2).

a(n) ~ 3^(3*n) / 2^(2*n+1) * (1 + 5/(3*sqrt(3*Pi*n))). (End)

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

a(n) = Sum_{k=0..n} 3^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*(9-4*g)) where g = 1+x*g^3 is the g.f. of A001764. - Seiichi Manyama, Aug 17 2025