cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A371778 a(n) = Sum_{k=0..floor(n/3)} binomial(3*n+2,n-3*k).

Original entry on oeis.org

1, 5, 28, 166, 1015, 6324, 39901, 254035, 1628380, 10493680, 67914088, 441086947, 2873255906, 18763759019, 122803467241, 805241108334, 5288922607095, 34789875710568, 229147231044397, 1511104857207706, 9975701630282920, 65920216186587257
Offset: 0

Views

Author

Seiichi Manyama, Apr 05 2024

Keywords

Crossrefs

Cf. A371777, A371779, A371780.
Cf. A066380.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(3*n+2, n-3*k));

Formula

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

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

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Apr 06 2024

Keywords

Crossrefs

Cf. A072547, A371814.
Cf. A005809, A066380, A165817, A385004, A386700.

Programs

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

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

A385823 a(n) = Sum_{k=0..n} binomial(3*n-3,k).

Original entry on oeis.org

1, 1, 7, 42, 256, 1586, 9949, 63004, 401930, 2579130, 16628809, 107636402, 699030226, 4552602248, 29722279084, 194458630304, 1274628824490, 8368726082346, 55027110808177, 362301656545966, 2388274575638228, 15760514137668514, 104108685843640517, 688331413734386356
Offset: 0

Views

Author

Seiichi Manyama, Aug 13 2025

Keywords

Links

Crossrefs

Cf. A066380, A160906, A386006, A387007, A387008, A387033.
Cf. A001764, A047099, A165817.

Programs

  • Magma
    [&+[Binomial(3*n-3,k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 27 2025
  • Mathematica
    Table[Sum[Binomial[3*n-3,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 27 2025 *)
  • PARI
    a(n) = sum(k=0, n, binomial(3*n-3, k));

Formula

a(n) = [x^n] (1+x)^(3*n-3)/(1-x).
a(n) = [x^n] 1/((1-x)^(2*n-3) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n-3,k) * binomial(3*n-k-4,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(3*n-k-4,n-k).
G.f.: 1/(g^2 * (2-g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764.

A386006 a(n) = Sum_{k=0..n} binomial(3*n-2,k).

Original entry on oeis.org

1, 2, 11, 64, 386, 2380, 14893, 94184, 600370, 3850756, 24821333, 160645504, 1043243132, 6794414896, 44360053772, 290244832992, 1902631226010, 12493030680180, 82153313341429, 540953389469312, 3566279609565226, 23536562549993228, 155489358646406149
Offset: 0

Views

Author

Seiichi Manyama, Aug 13 2025

Keywords

Links

Crossrefs

Cf. A066380, A160906, A385823, A387007, A387008, A387033.
Cf. A001764, A047099, A165817.

Programs

  • Magma
    [&+[Binomial(3*n-2,k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 27 2025
  • Mathematica
    Table[Sum[Binomial[3*n-2,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 27 2025 *)
  • PARI
    a(n) = sum(k=0, n, binomial(3*n-2, k));

Formula

a(n) = [x^n] (1+x)^(3*n-2)/(1-x).
a(n) = [x^n] 1/((1-x)^(2*n-2) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n-2,k) * binomial(3*n-k-3,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(3*n-k-3,n-k).
G.f.: 1/(g * (2-g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764.
Previous Showing 11-14 of 14 results.

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