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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A371816 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(3*n-3*k-1,n-3*k).

Original entry on oeis.org

1, 2, 10, 55, 322, 1947, 12013, 75154, 474946, 3024742, 19381045, 124797862, 806875421, 5234713031, 34060165282, 222174355575, 1452425614146, 9513309908589, 62418283102246, 410161124310550, 2698932409666237, 17781425199962255, 117281204608676426
Offset: 0

Views

Author

Seiichi Manyama, Apr 06 2024

Keywords

Crossrefs

Cf. A120305, A165817, A371817.
Cf. A371770.

Programs

  • Mathematica
    Table[Sum[(-1)^k Binomial[3n-3k-1,n-3k],{k,0,Floor[n/3]}],{n,0,30}] (* Harvey P. Dale, Aug 07 2025 *)
  • PARI
    a(n) = sum(k=0, n\3, (-1)^k*binomial(3*n-3*k-1, n-3*k));

Formula

a(n) = [x^n] 1/((1+x^3) * (1-x)^(2*n)).
a(n) = binomial(3*n-1, n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [1/3-n, 2/3-n, 1-n], -1). - Stefano Spezia, Apr 07 2024

A371817 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(4*n-3*k-1,n-3*k).

Original entry on oeis.org

1, 3, 21, 164, 1353, 11508, 99808, 877425, 7790745, 69704921, 627438606, 5675535000, 51546958296, 469764721533, 4293594852225, 39341599326304, 361271345551257, 3323924166943410, 30634431485945569, 282767849049333909, 2613630939017216898
Offset: 0

Views

Author

Seiichi Manyama, Apr 06 2024

Keywords

Crossrefs

Cf. A120305, A262977, A371816.
Cf. A371771.

Programs

  • PARI
    a(n) = sum(k=0, n\3, (-1)^k*binomial(4*n-3*k-1, n-3*k));

Formula

a(n) = [x^n] 1/((1+x^3) * (1-x)^(3*n)).
a(n) = binomial(4*n-1, n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [(1-4*n)/3, (2-4*n)/3, 1-4*n/3], -1). - Stefano Spezia, Apr 07 2024

A371837 a(n) = Sum_{k=0..floor(n/3)} n^k * binomial(2*n-3*k-1,n-3*k).

Original entry on oeis.org

1, 1, 3, 13, 51, 201, 834, 3529, 15075, 65431, 288278, 1285263, 5799470, 26492103, 122432628, 572291385, 2705760291, 12937116213, 62542367166, 305668511259, 1510080076410, 7539381024297, 38034307340076, 193835252945487, 997724306958606, 5185731234177001
Offset: 0

Views

Author

Seiichi Manyama, Apr 08 2024

Keywords

Crossrefs

Cf. A293574, A371836.
Cf. A120305, A371827.

Programs

  • Mathematica
    Join[{1}, Table[Sum[n^k*Binomial[2*n-3*k-1,n-1], {k, 0, n/3}], {n, 1, 25}]] (* Vaclav Kotesovec, Apr 08 2024 *)
  • PARI
    a(n) = sum(k=0, n\3, n^k*binomial(2*n-3*k-1, n-3*k));

Formula

a(n) = [x^n] 1/((1-n*x^3) * (1-x)^n).
a(n) ~ exp(n^(2/3) + n^(1/3)/2 + 1/3) * n^(n/3) / 3. - Vaclav Kotesovec, Apr 08 2024
Previous Showing 11-13 of 13 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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