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 1-2 of 2 results.

A190152 Triangle of binomial coefficients binomial(3*n-k,3*n-3*k).

Original entry on oeis.org

1, 1, 1, 1, 10, 1, 1, 28, 35, 1, 1, 55, 210, 84, 1, 1, 91, 715, 924, 165, 1, 1, 136, 1820, 5005, 3003, 286, 1, 1, 190, 3876, 18564, 24310, 8008, 455, 1, 1, 253, 7315, 54264, 125970, 92378, 18564, 680, 1, 1, 325, 12650, 134596, 490314, 646646, 293930, 38760, 969, 1
Offset: 0

Views

Author

Emanuele Munarini, May 05 2011

Keywords

Comments

From R. J. Mathar, Mar 15 2013: (Start)
The matrix inverse starts
1;
-1,1;
9,-10,1;
-288,322,-35,1;
22356,-25003,2730,-84,1;
-3428973,3835026,-418825,12936,-165,1;
914976405,-1023326973,111759115,-3452449,44187,-286,1;
... (End)

Examples

			Triangle begins:
  1
  1, 1
  1, 10, 1
  1, 28, 35, 1
  1, 55, 210, 84, 1
  1, 91, 715, 924, 165, 1
  1, 136, 1820, 5005, 3003, 286, 1
  1, 190, 3876, 18564, 24310, 8008, 455, 1
  1, 253, 7315, 54264, 125970, 92378, 18564, 680, 1
  ...

Links

Crossrefs

Cf. A000447 (first subdiagonal), A053135 (second subdiagonal), A060544 (second column), A190088, A190153 (row sums), A190154 (diagonal sums).

Programs

  • Mathematica
    Flatten[Table[Binomial[3n - k, 3n - 3k], {n, 0, 9}, {k, 0, n}]]
  • Maxima
    create_list(binomial(3*n-k,3*n-3*k),n,0,9,k,0,n);
  • PARI
    for(n=0,10, for(k=0,n, print1(binomial(3*n-k, 3*(n-k)), ", "))) \\ G. C. Greubel, Dec 29 2017

A190153 Row sums of the triangle A190152.

Original entry on oeis.org

1, 2, 12, 65, 351, 1897, 10252, 55405, 299426, 1618192, 8745217, 47261895, 255418101, 1380359512, 7459895657, 40315615410, 217878227876, 1177482265857, 6363483400447, 34390259761825, 185855747875876, 1004422742303477, 5428215467030962
Offset: 0

Views

Author

Emanuele Munarini, May 05 2011

Keywords

Links

Crossrefs

Cf. A190152, A190154.

Programs

  • Magma
    I:=[1, 2, 12]; [n le 3 select I[n] else 5*Self(n-1) +2*Self(n-2) + Self(n-3): n in [1..30]]; // G. C. Greubel, Dec 30 2017
  • Maple
    seq(add(binomial(3*n-k,3*n-3*k), k=0..n), n=0..20);
  • Mathematica
    Table[Sum[Binomial[3n - k, 3n - 3k], {k, 0, n}], {n, 0, 22}]
LinearRecurrence[{5,2,1}, {1,2,12}, 30] (* G. C. Greubel, Dec 30 2017 *)
  • Maxima
    makelist(sum(binomial(3*n-k,3*n-3*k),k,0,n),n,0,22);
  • PARI
    x='x+O('x^30); Vec((1-3*x)/(1-5*x-2*x^2-x^3)) \\ G. C. Greubel, Dec 30 2017

Formula

a(n) = Sum_{k=0..n} binomial(3*n-k,3*n-3*k).
From Colin Barker, Mar 21 2012: (Start)
a(n) = 5*a(n-1) + 2*a(n-2) + a(n-3).
G.f.: (1-3*x)/(1-5*x-2*x^2-x^3). (End)
Showing 1-2 of 2 results.

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

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