A371813 -id:A371813 - cp's OEIS frontend

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

Seiichi Manyama, Jul 30 2025

Cf. A000244, A100192, A385498, A386699.
Cf. A005809, A066380, A165817, A371813, A386700.
Cf. A001764, A386617.

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 *)

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

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

A386700 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(3*n,k).

Original entry on oeis.org

1, 0, 6, 30, 186, 1140, 7116, 44856, 285066, 1823232, 11721726, 75683718, 490429224, 3187723344, 20774505408, 135699314640, 888177411018, 5823660624408, 38245666664994, 251528316024042, 1656338630258826, 10919849458481028, 72068276593960884, 476093333668519872

Offset: 0

Seiichi Manyama, Jul 30 2025

Cf. A122803, A386701, A386702.

Cf. A005809, A066380, A165817, A371813, A385004.

Table[(-8/9)^n - Binomial[3*n, n]*(-1 + Hypergeometric2F1[1, -2*n, 1 + n, 1/3]), {n, 0, 25}] (* Vaclav Kotesovec, Jul 30 2025 *)

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

a(n) = [x^n] 1/((1+2*x) * (1-x)^(2*n)).
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(2*n+k-1,k).
From Vaclav Kotesovec, Jul 30 2025: (Start)
Recurrence: 18*n*(2*n - 1)*(55*n^2 - 175*n + 138)*a(n) = (11605*n^4 - 49410*n^3 + 74243*n^2 - 46014*n + 9720)*a(n-1) + 24*(3*n - 5)*(3*n - 4)*(55*n^2 - 65*n + 18)*a(n-2).
a(n) ~ 3^(3*n + 1/2) / (5 * sqrt(Pi*n) * 2^(2*n)). (End)
G.f.: g/((-2+3*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} (-2)^k * 3^(n-k) * binomial(3*n,k) * binomial(3*n-k-1,n-k). - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - 6*x*g^2*(-1+g)) where g = 1+x*g^3 is the g.f. of A001764. - Seiichi Manyama, Aug 17 2025

A371814 a(n) = Sum_{k=0..n} (-1)^k * binomial(4*n-k-1,n-k).

Original entry on oeis.org

1, 2, 16, 128, 1068, 9142, 79612, 701864, 6244892, 55962920, 504375396, 4567003520, 41513817444, 378596616452, 3462411408136, 31742042431048, 291616814436124, 2684123914512280, 24746511514749280, 228491677484832896, 2112549277665243328

Offset: 0

Seiichi Manyama, Apr 06 2024

nonn

Cf. A072547, A371813.
Cf. A005810, A262977.
Cf. A066381, A262977, A385498, A386701.

a(n) = sum(k=0, n, (-1)^k*binomial(4*n-k-1, n-k));

a(n) = [x^n] 1/((1+x) * (1-x)^(3*n)).
a(n) = binomial(4*n-1, n)*hypergeom([1, -n], [1-4*n], -1). - Stefano Spezia, Apr 07 2024
From Vaclav Kotesovec, Apr 07 2024: (Start)
Recurrence: 24*n*(3*n - 2)*(3*n - 1)*(415*n^3 - 1898*n^2 + 2871*n - 1436)*a(n) = (838715*n^6 - 5099533*n^5 + 12225995*n^4 - 14652035*n^3 + 9157250*n^2 - 2799192*n + 322560)*a(n-1) + 8*(2*n - 3)*(4*n - 7)*(4*n - 5)*(415*n^3 - 653*n^2 + 320*n - 48)*a(n-2).
a(n) ~ 2^(8*n + 1/2) / (5 * sqrt(Pi*n) * 3^(3*n - 1/2)). (End)
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(4*n,k). - Seiichi Manyama, Jul 30 2025
G.f.: g/((-1+2*g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 13 2025
a(n) = Sum_{k=0..n} (-1)^k * 2^(n-k) * binomial(4*n,k) * binomial(4*n-k-1,n-k). - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g^3*(-4+6*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 17 2025

cp's OEIS Frontend

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

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A386700 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(3*n,k).

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A371814 a(n) = Sum_{k=0..n} (-1)^k * binomial(4*n-k-1,n-k).

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