A141723 Triangle T(n, k) = Sum_{j=0..n} (2*n)!/((2*n-k-j)!*j!*k!), read by rows.

1, 3, 4, 11, 28, 24, 42, 156, 225, 160, 163, 792, 1596, 1736, 1120, 638, 3820, 9855, 14400, 13230, 8064, 2510, 17832, 55968, 102520, 122265, 100584, 59136, 9908, 81368, 300482, 661024, 968968, 1005004, 765765, 439296, 39203, 365104, 1549320, 3975440, 6910540, 8653008, 8112104, 5845840, 3294720

Offset

0,2

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n, k) = Sum_{j=0..n} (2*n)!/((2*n-k-j)!*j!*k!).

Example

Triangle begins as:
     1;
     3,     4;
    11,    28,     24;
    42,   156,    225,    160;
   163,   792,   1596,   1736,   1120;
   638,  3820,   9855,  14400,  13230,   8064;
  2510, 17832,  55968, 102520, 122265,  100584,  59136;
  9908, 81368, 300482, 661024, 968968, 1005004, 765765, 439296;

Mathematica

Table[Sum[Multinomial[2*n-k-j, k, j], {j, 0, n}], {n, 0, 12}, {k, 0, n}]//Flatten

Prog

(Magma) F:= Factorial; [(&+[F(2*n)/(F(k)*F(j)*F(2*n-k-j)): j in [0..n]]): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 28 2021
(Sage) f=factorial; flatten([[sum(f(2*n)/(f(k)*f(j)*f(2*n-k-j)) for j in (0..n)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 28 2021

Crossrefs

Sequence in context: A198443 A041231 A042129 * A268478 A180363 A100845

Adjacent sequences: A141720 A141721 A141722 * A141724 A141725 A141726

Keyword

nonn,tabl

Author

Roger L. Bagula, Sep 12 2008

Extensions

Edited by G. C. Greubel, Mar 28 2021

Status

approved