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 1-8 of 8 results.

A377148 a(n) = Sum_{k=0..n} binomial(k+3,3) * binomial(k,n-k)^2.

Original entry on oeis.org

1, 4, 14, 60, 225, 796, 2764, 9304, 30580, 98700, 313422, 981548, 3037473, 9301620, 28222000, 84927760, 253699285, 752863840, 2220831160, 6515581600, 19021079866, 55276625304, 159967084164, 461150383400, 1324652146775, 3792447499916, 10824189204014
Offset: 0

Views

Author

Seiichi Manyama, Oct 18 2024

Keywords

Links

Crossrefs

Cf. A051286, A182884, A377145, A377152, A377153, A377158, A377159.
Cf. A377149, A377150.
Cf. A089627.

Programs

  • Magma
    [&+[Binomial(k+3,3)*Binomial(k, n-k)^2: k in [0..n]]: n in [0..30]]; // Vincenzo Librandi, May 12 2025
  • Mathematica
    Table[Sum[Binomial[k+3,3]*Binomial[k, n-k]^2,{k,0,n}],{n,0,30}] (* Vincenzo Librandi, May 12 2025 *)
  • PARI
    a(n) = sum(k=0, n, binomial(k+3, 3)*binomial(k, n-k)^2);
  • PARI
    a089627(n, k) = n!/((n-2*k)!*k!^2);
my(N=3, 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) * ((1-x-x^2)^2 + 6*x^3) / ((1-x-x^2)^2 - 4*x^3)^(7/2).

A382230 a(n) = Sum_{k=0..n} binomial(k+2,2) * binomial(2*k,2*n-2*k).

Original entry on oeis.org

1, 3, 9, 46, 171, 591, 2033, 6714, 21606, 68308, 212370, 651234, 1974113, 5924277, 17623671, 52025858, 152539077, 444530073, 1288396257, 3715833732, 10668907932, 30507914696, 86912853588, 246755125332, 698353551105, 1970673504951, 5545952371509, 15568330002486
Offset: 0

Views

Author

Seiichi Manyama, Mar 28 2025

Keywords

Links

Crossrefs

Cf. A108479, A381421, A382470, A382471, A382472, A382473, A382474.
Cf. A034839, A377145.

Programs

  • Magma
    [&+[Binomial(k+2, 2) * Binomial(2*k, 2*n-2*k): k in [0..n]]: n in [0..29]]; // Vincenzo Librandi, Apr 22 2025
  • Mathematica
    Table[Sum[Binomial[k+2,2]*Binomial[2*k,2*n-2*k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Apr 22 2025 *)
  • PARI
    a(n) = sum(k=0, n, binomial(k+2, 2)*binomial(2*k, 2*n-2*k));
  • PARI
    my(N=2, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); 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-x^2)^(3-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^3.
a(n) = 6*a(n-1) - 9*a(n-2) + 2*a(n-3) - 18*a(n-4) + 30*a(n-5) + 7*a(n-6) + 30*a(n-7) - 18*a(n-8) + 2*a(n-9) - 9*a(n-10) + 6*a(n-11) - a(n-12).

A377152 a(n) = Sum_{k=0..n} binomial(k+4,4) * binomial(k,n-k)^2.

Original entry on oeis.org

1, 5, 20, 95, 400, 1561, 5915, 21610, 76585, 265075, 898622, 2992235, 9810290, 31727815, 101379175, 320464280, 1003259080, 3113576320, 9586763720, 29305985800, 88997753446, 268642069750, 806394498200, 2408144329250, 7157177344225, 21177323087891
Offset: 0

Views

Author

Seiichi Manyama, Oct 18 2024

Keywords

Links

Crossrefs

Cf. A051286, A182884, A377145, A377148, A377153, A377158, A377159.
Cf. A001873, A089627.

Programs

  • Maple
    f:= proc(n) local k; add(binomial(k+4,4)*binomial(k,n-k)^2,k=0..n) end proc:
map(f, [$0..50]); # Robert Israel, Dec 05 2024
  • PARI
    a(n) = sum(k=0, n, binomial(k+4, 4)*binomial(k, n-k)^2);
  • PARI
    a089627(n, k) = n!/((n-2*k)!*k!^2);
my(N=4, 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.: (Sum_{k=0..2} A089627(4,k) * (1-x-x^2)^(4-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^(9/2).

A377153 a(n) = Sum_{k=0..n} binomial(k+5,5) * binomial(k,n-k)^2.

Original entry on oeis.org

1, 6, 27, 140, 651, 2772, 11354, 44640, 169371, 624742, 2248575, 7922124, 27397937, 93214632, 312559200, 1034507696, 3384194616, 10954244952, 35118346760, 111602517096, 351819819414, 1100912299156, 3421515852834, 10566654790176, 32441857824859, 99060134392422
Offset: 0

Views

Author

Seiichi Manyama, Oct 18 2024

Keywords

Crossrefs

Cf. A051286, A182884, A377145, A377148, A377152, A377158, A377159.
Cf. A001874, A089627.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(k+5, 5)*binomial(k, n-k)^2);
  • PARI
    a089627(n, k) = n!/((n-2*k)!*k!^2);
my(N=5, 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.: (Sum_{k=0..2} A089627(5,k) * (1-x-x^2)^(5-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^(11/2).

A377158 a(n) = Sum_{k=0..n} binomial(k+6,6) * binomial(k,n-k)^2.

Original entry on oeis.org

1, 7, 35, 196, 994, 4578, 20118, 84540, 341397, 1335103, 5078227, 18852428, 68519920, 244413820, 857393700, 2963013816, 10102413972, 34025396580, 113329367816, 373642488044, 1220412680410, 3951964394642, 12695738508950, 40484919514284, 128216539026261
Offset: 0

Views

Author

Seiichi Manyama, Oct 18 2024

Keywords

Crossrefs

Cf. A051286, A182884, A377145, A377148, A377152, A377153, A377159.
Cf. A089627.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(k+6, 6)*binomial(k, n-k)^2);
  • PARI
    a089627(n, k) = n!/((n-2*k)!*k!^2);
my(N=6, 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.: (Sum_{k=0..3} A089627(6,k) * (1-x-x^2)^(6-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^(13/2).

A377159 a(n) = Sum_{k=0..n} binomial(k+7,7) * binomial(k,n-k)^2.

Original entry on oeis.org

1, 8, 44, 264, 1446, 7152, 33516, 149688, 640233, 2642992, 10582220, 41249000, 157050660, 585621960, 2143442400, 7715164176, 27353809188, 95660348904, 330377130644, 1127996393656, 3810881349814, 12750188169312, 42276102419916, 139008143200536, 453526927536969
Offset: 0

Views

Author

Seiichi Manyama, Oct 18 2024

Keywords

Crossrefs

Cf. A051286, A182884, A377145, A377148, A377152, A377153, A377158.
Cf. A089627.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(k+7, 7)*binomial(k, n-k)^2);
  • PARI
    a089627(n, k) = n!/((n-2*k)!*k!^2);
my(N=7, 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.: (Sum_{k=0..3} A089627(7,k) * (1-x-x^2)^(7-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^(15/2).

A377146 a(n) = Sum_{k=0..floor(n/2)} binomial(k+2,2) * binomial(k,n-2*k)^2.

Original entry on oeis.org

1, 0, 3, 3, 6, 24, 16, 90, 105, 250, 561, 765, 2143, 3108, 6861, 12985, 22221, 47988, 79463, 161451, 293610, 535836, 1042188, 1835898, 3534766, 6399198, 11805756, 22021232, 39718497, 74193924, 134489713, 247165839, 453235266, 822748406, 1512078192, 2741606052
Offset: 0

Views

Author

Seiichi Manyama, Oct 17 2024

Keywords

Crossrefs

Cf. A377145, A377147.
Cf. A298567, A375218.
Cf. A089627.

Programs

  • PARI
    a(n) = sum(k=0, n\2, binomial(k+2, 2)*binomial(k, n-2*k)^2);
  • PARI
    a089627(n, k) = n!/((n-2*k)!*k!^2);
my(N=2, M=40, x='x+O('x^M), X=1-x^2-x^3, Y=5); 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^2-x^3)^2 + 2*x^5) / ((1-x^2-x^3)^2 - 4*x^5)^(5/2).

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

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 1-8 of 8 results.

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

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