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.

A377145 a(n) = Sum_{k=0..n} binomial(k+2,2) * binomial(k,n-k)^2.

Original entry on oeis.org

1, 3, 9, 34, 111, 351, 1103, 3384, 10224, 30536, 90222, 264186, 767663, 2215623, 6356907, 18143300, 51540885, 145801395, 410888595, 1153964520, 3230723826, 9019081038, 25112021154, 69750583164, 193303849531, 534602071341, 1475644537323, 4065845732794
Offset: 0

Views

Author

Seiichi Manyama, Oct 17 2024

Keywords

Crossrefs

Cf. A051286, A182884, A377148, A377152, A377153, A377158, A377159.
Cf. A377146, A377147.
Cf. A089627.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(k+2, 2)*binomial(k, n-k)^2);
  • PARI
    a089627(n, k) = n!/((n-2*k)!*k!^2);
my(N=2, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); 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-x^2)^2 + 2*x^3) / ((1-x-x^2)^2 - 4*x^3)^(5/2).

A382494 a(n) = Sum_{k=0..floor(n/2)} binomial(k+2,2) * binomial(2*k,2*n-4*k).

Original entry on oeis.org

1, 0, 3, 3, 6, 36, 16, 150, 165, 430, 1071, 1365, 4453, 6258, 14841, 29169, 49941, 115356, 190091, 404811, 750792, 1393956, 2808438, 4988268, 9905746, 18207126, 34231566, 65278964, 119255889, 227648406, 418394087, 782045001, 1457704212, 2681909302
Offset: 0

Views

Author

Seiichi Manyama, Mar 29 2025

Keywords

Links

Crossrefs

Cf. A376729, A382300, A382495.
Cf. A034839, A377146, A382230.

Programs

  • Magma
    [&+[Binomial(k+2, 2)*Binomial(2*k, 2*n-4*k): k in [0..n]]: n in [0..41]]; // Vincenzo Librandi, May 11 2025
  • Mathematica
    Table[Sum[Binomial[k+2,2]*Binomial[2*k, 2*n-4*k],{k,0,Floor[n/2]}],{n,0,30}] (* Vincenzo Librandi, May 11 2025 *)
  • PARI
    a(n) = sum(k=0, n\2, binomial(k+2, 2)*binomial(2*k, 2*n-4*k));
  • PARI
    my(N=2, M=40, x='x+O('x^M), X=1-x^2-x^3, Y=5); Vec(sum(k=0, (N+1)\2, 4^k*binomial(N+1, 2*k)*X^(N+1-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1))

Formula

G.f.: (Sum_{k=0..1} 4^k * binomial(3,2*k) * (1-x^2-x^3)^(3-2*k) * x^(5*k)) / ((1-x^2-x^3)^2 - 4*x^5)^3.
a(n) = 6*a(n-2) + 6*a(n-3) - 15*a(n-4) - 18*a(n-5) + 5*a(n-6) + 12*a(n-7) - 3*a(n-8) + 32*a(n-9) + 12*a(n-10) - 6*a(n-11) - 4*a(n-12) + 18*a(n-13) - 33*a(n-14) + 26*a(n-15) - 15*a(n-16) + 6*a(n-17) - a(n-18).

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

Cf. A377145, A377146.
Cf. A375470, A376721.
Cf. A089627.

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.

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