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.

Showing 1-4 of 4 results.

A387007 a(n) = Sum_{k=0..n} binomial(3*n+2,k).

Original entry on oeis.org

1, 6, 37, 232, 1471, 9402, 60460, 390656, 2533987, 16489546, 107594213, 703680424, 4611412196, 30273024984, 199045392232, 1310535994368, 8639411571051, 57017083602138, 376674527189599, 2490742704227192, 16483857933928471, 109175823528400778, 723611538997758784
Offset: 0

Views

Author

Seiichi Manyama, Aug 12 2025

Keywords

Links

Crossrefs

Cf. A066380, A160906, A387008.
Cf. A001764.

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+1,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(3*n-k+1,n-k).
G.f.: g^3/((2-g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764.
D-finite with recurrence: 24*(5*n+1)*(3*n-1)*(3*n-2)*a(n-2) -(295*n^3-156*n^2-61*n+6)*a(n-1) +2*n*(2*n+1)*(5*n-4)*a(n). - Georg Fischer, Aug 17 2025

A387033 a(n) = Sum_{k=0..n} binomial(3*n-1,k).

Original entry on oeis.org

1, 3, 16, 93, 562, 3473, 21778, 137980, 880970, 5658537, 36519556, 236618693, 1538132224, 10026362492, 65513177704, 428957009288, 2813768603466, 18486790962201, 121634649321208, 801330506737399, 5285305708097522, 34896814868837161, 230631268849574378
Offset: 0

Views

Author

Seiichi Manyama, Aug 13 2025

Keywords

Links

Crossrefs

Cf. A066380, A160906, A385823, A386006, A387007, A387008.
Cf. A114121, A387037.
Cf. A001764, A047099, A165817.

Programs

  • Magma
    [&+[Binomial(3*n-1,k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 27 2025
  • Mathematica
    Table[Sum[Binomial[3*n-1,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 27 2025 *)
  • PARI
    a(n) = sum(k=0, n, binomial(3*n-1, k));

Formula

a(n) = [x^n] (1+x)^(3*n-1)/(1-x).
a(n) = [x^n] 1/((1-x)^(2*n-1) * (1-2*x)).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n-1,k) * binomial(3*n-k-2,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(3*n-k-2,n-k).
G.f.: 1/((2-g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764.
D-finite with recurrence: 24*(5*n+6)*(3*n-4)*(3*n-5)*a(n-2)-(295*n^3-451*n^2-234*n+360)*a(n-1)+2*n*(5*n+1)*(2*n-3)*a(n) = 0. - Georg Fischer, Aug 17 2025
a(n) ~ 3^(3*n - 1/2) / (sqrt(Pi*n) * 2^(2*n-1)). - Vaclav Kotesovec, Aug 27 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.
Showing 1-4 of 4 results.

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