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.

A190088 Triangle of binomial coefficients binomial(3*n-k+1,3*n-3*k+1).

Original entry on oeis.org

1, 1, 3, 1, 15, 5, 1, 36, 70, 7, 1, 66, 330, 210, 9, 1, 105, 1001, 1716, 495, 11, 1, 153, 2380, 8008, 6435, 1001, 13, 1, 210, 4845, 27132, 43758, 19448, 1820, 15, 1, 276, 8855, 74613, 203490, 184756, 50388, 3060, 17, 1, 351, 14950, 177100, 735471, 1144066, 646646, 116280, 4845, 19
Offset: 0

Views

Author

Emanuele Munarini, May 04 2011

Keywords

Comments

Row sums = A190089.
Diagonal sums = A190090.

Examples

			Triangle begins:
1
1, 3
1, 15, 5
1, 36, 70, 7
1, 66, 330, 210, 9
1, 105, 1001, 1716, 495, 11
1, 153, 2380, 8008, 6435, 1001, 13
1, 210, 4845, 27132, 43758, 19448, 1820, 15
1, 276, 8855, 74613, 203490, 184756, 50388, 3060, 17

Links

Crossrefs

Cf. A190089, A190090.

Programs

  • Magma
    /* As triangle */ [[Binomial(3*n-k+1, 3*n-3*k+1): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Mar 04 2018
  • Mathematica
    Flatten[Table[Binomial[3n - k + 1, 3n - 3k + 1], {n, 0, 8}, {k, 0, n}]]
  • Maxima
    create_list(binomial(3*n-k+1,3*n-3*k+1),n,0,12,k,0,n);
  • PARI
    for(n=0,10, for(k=0,n, print1(binomial(3*n-k+1, 3*n-3*k+1), ", "))) \\ G. C. Greubel, Mar 04 2018

A190090 Diagonal sums of the triangular matrix A190088.

Original entry on oeis.org

1, 1, 4, 16, 42, 137, 443, 1365, 4316, 13625, 42785, 134758, 424331, 1335378, 4203927, 13233947, 41657808, 131135696, 412803240, 1299458257, 4090567673, 12876698159, 40534529294, 127598621869, 401667591501, 1264408966284, 3980231826575, 12529367967276, 39441185140197
Offset: 0

Views

Author

Emanuele Munarini, May 04 2011

Keywords

Links

Crossrefs

Cf. A190088, A190089.

Programs

  • Magma
    [(&+[Binomial(3*n-4*k+1,3*n-6*k+1): k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Mar 04 2018
  • Mathematica
    Table[Sum[Binomial[3n - 4k + 1, 3n - 6k + 1], {k, 0, n/2}], {n, 0, 26}]
LinearRecurrence[{2,2,6,-3,0,1},{1,1,4,16,42,137},27] (* Harvey P. Dale, Jul 04 2011 *)
  • Maxima
    makelist(sum(binomial(3*n-4*k+1,3*n-6*k+1),k,0,n/2),n,0,12);
  • PARI
    Vec((1-x-x^4)/(1-2*x-2*x^2-6*x^3+3*x^4-x^6)+O(x^29)) \\ Charles R Greathouse IV, Jun 30 2011

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-4*k+1,3*n-6*k+1).
G.f.: (1-x-x^4)/(1-2*x-2*x^2-6*x^3+3*x^4-x^6).
a(n) = 2*a(n-1)+ 2*a(n-2)+ 6*a(n-3)-3*a(n-4)+a(n-6), and a(0)=1, a(1)=1, a(2)=4, a(3)=16, a(4)=42, a(5)=137, . - Harvey P. Dale, Jul 04 2011
Showing 1-2 of 2 results.

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

Content is available under The OEIS End-User License Agreement.