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 1-3 of 3 results.

A375470 a(n) = Sum_{k=0..floor(n/3)} (k+1) * binomial(k,n-3*k)^2.

Original entry on oeis.org

1, 0, 0, 2, 2, 0, 3, 12, 3, 4, 36, 36, 9, 80, 180, 86, 155, 600, 607, 402, 1581, 2808, 1967, 3780, 9816, 10376, 10584, 28626, 44918, 41184, 77627, 160436, 181044, 228972, 499512, 735654, 811823, 1467072, 2640231, 3191642, 4494502, 8566308, 12280547, 15315974, 26498718
Offset: 0

Views

Author

Seiichi Manyama, Oct 17 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = sum(k=0, n\3, (k+1)*binomial(k, n-3*k)^2);

Formula

G.f.: (1-x^3-x^4)/((1-x^3-x^4)^2 - 4*x^7)^(3/2).

A376722 Expansion of 1/sqrt((1 - x^4 - x^5)^2 - 4*x^9).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 0, 0, 1, 4, 1, 0, 1, 9, 9, 1, 1, 16, 36, 16, 2, 25, 100, 100, 26, 37, 225, 400, 226, 85, 442, 1225, 1226, 505, 833, 3137, 4901, 3217, 2080, 7120, 15878, 15976, 9081, 15696, 44182, 63626, 47125, 41625, 110926, 213688, 217801, 157300, 272251, 630458
Offset: 0

Views

Author

Seiichi Manyama, Oct 02 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=60, x='x+O('x^N)); Vec(1/sqrt((1-x^4-x^5)^2-4*x^9))
    
  • PARI
    a(n) = sum(k=0, n\4, binomial(k, n-4*k)^2);

Formula

a(n) = Sum_{k=0..floor(n/4)} binomial(k,n-4*k)^2.

A377147 a(n) = Sum_{k=0..floor(n/3)} binomial(k+2,2) * binomial(k,n-3*k)^2.

Original entry on oeis.org

1, 0, 0, 3, 3, 0, 6, 24, 6, 10, 90, 90, 25, 240, 540, 261, 540, 2100, 2128, 1533, 6321, 11236, 8064, 16884, 44173, 46980, 51156, 142939, 224991, 212400, 423426, 882660, 1006875, 1338558, 2991318, 4431669, 5034296, 9457704, 17178678, 21059737, 30809286, 59843394, 86518266
Offset: 0

Views

Author

Seiichi Manyama, Oct 17 2024

Keywords

Crossrefs

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(k+2, 2)*binomial(k, n-3*k)^2);
    
  • PARI
    a089627(n, k) = n!/((n-2*k)!*k!^2);
    my(N=2, M=50, x='x+O('x^M), X=1-x^3-x^4, Y=7); Vec(sum(k=0, N\2, a089627(N, k)*X^(N-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1/2))

Formula

G.f.: ((1-x^3-x^4)^2 + 2*x^7) / ((1-x^3-x^4)^2 - 4*x^7)^(5/2).
Showing 1-3 of 3 results.