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.

A178820 Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.

Original entry on oeis.org

1, 4, 4, 10, 20, 10, 20, 60, 60, 20, 35, 140, 210, 140, 35, 56, 280, 560, 560, 280, 56, 84, 504, 1260, 1680, 1260, 504, 84, 120, 840, 2520, 4200, 4200, 2520, 840, 120, 165, 1320, 4620, 9240, 11550, 9240, 4620, 1320, 165, 220, 1980, 7920, 18480, 27720, 27720, 18480, 7920, 1980, 220
Offset: 0

Views

Author

Harlan J. Brothers, Jun 17 2010

Keywords

Comments

The product of the tetrahedral numbers (A000292, beginning with second term) and Pascal's triangle (A007318). Also level 4 of Pascal's prism (A178819): (i+3; 3, i-j, j), i >= 0, 0 <= j <= i.

Examples

			Triangle begins:
   1;
   4,   4;
  10,  20,  10;
  20,  60,  60,  20;
  35, 140, 210, 140,  35;

Links

Crossrefs

Cf. A000292, A007318, A178819.
Rows sums give A001789.

Programs

  • GAP
    T:=Flat(List([0..10], n-> List([0..n], k-> Binomial(n+3, 3)* Binomial(n, k) ))); # G. C. Greubel, Jan 22 2019
  • Magma
    /* As triangle */ [[Binomial(n+3,3)*Binomial(n,k): k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Oct 23 2017
  • Maple
    T:=(n,k)->binomial(n+3,3)*binomial(n,k): seq(seq(T(n,k),k=0..n),n=0..9); # Muniru A Asiru, Jan 22 2019
  • Mathematica
    Table[Multinomial[3, i-j, j], {i, 0, 9}, {j, 0, i}]//Column
  • PARI
    {T(n,k) = binomial(n+3, 3)*binomial(n, k)}; \\ G. C. Greubel, Jan 22 2019
  • Sage
    [[binomial(n+3, 3)*binomial(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jan 22 2019

Formula

T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.
For element a in A178819: a_(4, i, j) = (i+2; 3, i-j, j-1), i >= 1, 1 <= j <= i.
G.f.: 1/(1 - x - x*y)^4. - Ilya Gutkovskiy, Mar 20 2020

A178821 Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.

Original entry on oeis.org

1, 5, 5, 15, 30, 15, 35, 105, 105, 35, 70, 280, 420, 280, 70, 126, 630, 1260, 1260, 630, 126, 210, 1260, 3150, 4200, 3150, 1260, 210, 330, 2310, 6930, 11550, 11550, 6930, 2310, 330, 495, 3960, 13860, 27720, 34650, 27720, 13860, 3960, 495, 715, 6435, 25740, 60060, 90090, 90090, 60060, 25740, 6435, 715
Offset: 0

Views

Author

Harlan J. Brothers, Jun 19 2010

Keywords

Comments

The product of the pentatope numbers (A000332, beginning with fifth term) and Pascal's triangle (A007318). Also level 5 of Pascal's prism (A178819) read by rows: (i+4; 4, i-j, j), i >= 0, 0 <= j <= i.

Examples

			Triangle begins:
   1;
   5,   5;
  15,  30,  15;
  35, 105, 105,  35;
  70, 280, 420, 280,  70;

Links

Crossrefs

Cf. A000332, A007318, A178819, A178820, A178822.
Rows sum to A003472, shallow diagonals sum to A001873.

Programs

  • GAP
    T:=Flat(List([0..10], n-> List([0..n], k-> Binomial(n+4, 4)* Binomial(n, k) ))); # G. C. Greubel, Jan 22 2019
  • Magma
    /* As triangle */ [[Binomial(n+4,4)*Binomial(n,k): k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Oct 23 2017
  • Maple
    T:=(n,k)->binomial(n+4,4)*binomial(n,k): seq(seq(T(n,k),k=0..n),n=0..9); # Muniru A Asiru, Jan 22 2019
  • Mathematica
    Table[Multinomial[4, i-j, j], {i, 0, 9}, {j, 0, i}]//Column
  • PARI
    {T(n,k) = binomial(n+4, 4)*binomial(n, k)}; \\ G. C. Greubel, Jan 22 2019
  • Sage
    [[binomial(n+4, 4)*binomial(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jan 22 2019

Formula

T(n,k) = C(n+4,4) * C(n,k), 0 <= k <= n.
For element a in A178819: a_(5, i, j) = (i+3; 4, i-j, j-1), i >= 1, 1 <= j <= i.
G.f.: 1/(1 - x - x*y)^5. - Ilya Gutkovskiy, Mar 20 2020

A178822 Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.

Original entry on oeis.org

1, 6, 6, 21, 42, 21, 56, 168, 168, 56, 126, 504, 756, 504, 126, 252, 1260, 2520, 2520, 1260, 252, 462, 2772, 6930, 9240, 6930, 2772, 462, 792, 5544, 16632, 27720, 27720, 16632, 5544, 792, 1287, 10296, 36036, 72072, 90090, 72072, 36036, 10296, 1287
Offset: 0

Views

Author

Harlan J. Brothers, Jun 19 2010

Keywords

Comments

The product of A000389 and Pascal's triangle (A007318). Level 6 of Pascal's prism (A178819) read by rows: (i+5; 5, i-j, j), i >= 0, 0 <= j <= i.

Examples

			Triangle begins:
    1;
    6,   6;
   21,  42,  21;
   56, 168, 168,  56;
  126, 504, 756, 504, 126;

Links

Crossrefs

Cf. A000389, A007318, A178819, A178820, A178821.
Rows sum to A054849, shallow diagonals sum to A001874.

Programs

  • Magma
    /* As triangle */ [[Binomial(n+5,5)*Binomial(n,k): k in [0..n]]: n in [0..10]]; // Vincenzo Librandi, Oct 23 2017
  • Mathematica
    Table[Multinomial[5, i-j, j], {i, 0, 9}, {j, 0, i}]//Column
Table[Binomial[n + 5, 5]*Binomial[n, k], {n,0,10}, {k,0,n}] // Flatten (* G. C. Greubel, Nov 25 2017 *)
  • PARI
    for(n=0,10, for(k=0,n, print1(binomial(n+5,5)*binomial(n,k), ", "))) \\ G. C. Greubel, Nov 25 2017

Formula

T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.
For element a_(h, i, j) in A178819: a_(6, i, j) = (i+4; 5, i-j, j-1), i >= 1, 1 <= j <= i.
G.f.: 1/(1 - x - x*y)^6. - Ilya Gutkovskiy, Mar 20 2020
Showing 1-3 of 3 results.

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