A171822 Triangle T(n,k) = binomial(2*n-k, k)*binomial(n+k, 2*k), read by rows.

1, 1, 1, 1, 9, 1, 1, 30, 30, 1, 1, 70, 225, 70, 1, 1, 135, 980, 980, 135, 1, 1, 231, 3150, 7056, 3150, 231, 1, 1, 364, 8316, 34650, 34650, 8316, 364, 1, 1, 540, 19110, 132132, 245025, 132132, 19110, 540, 1, 1, 765, 39600, 420420, 1288287, 1288287, 420420, 39600, 765, 1

Offset

0,5

Links

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Formula

T(n, k) = binomial(2*n-k, k)*binomial(n+k, 2*k) = A054142(n, k)*A085478(n, k).

Sum_{k=0..n} T(n, k) = Hypergeometric 4F3([-n, -n, 1/2 -n, n+1], [1/2, 1, -2*n], 1) = A183160(n). - G. C. Greubel, Feb 22 2021

Example

Triangle begins as:
  1;
  1,    1;
  1,    9,     1;
  1,   30,    30,       1;
  1,   70,   225,      70,       1;
  1,  135,   980,     980,     135,       1;
  1,  231,  3150,    7056,    3150,     231,       1;
  1,  364,  8316,   34650,   34650,    8316,     364,       1;
  1,  540, 19110,  132132,  245025,  132132,   19110,     540,     1;
  1,  765, 39600,  420420, 1288287, 1288287,  420420,   39600,   765,    1;
  1, 1045, 75735, 1166880, 5465460, 9018009, 5465460, 1166880, 75735, 1045, 1;

Mathematica

Table[Binomial[2*n-k, k]*Binomial[n+k, 2*k], {n, 0, 10}, {k, 0, n}]//Flatten

Prog

(Sage) flatten([[binomial(2*n-k, k)*binomial(n+k, 2*k) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Feb 22 2021
(Magma) [Binomial(2*n-k, k)*Binomial(n+k, 2*k): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 22 2021

Crossrefs

Cf. A054142, A085478, A183160.

Sequence in context: A144404 A014761 A073702 * A176490 A370232 A174158

Adjacent sequences: A171819 A171820 A171821 * A171823 A171824 A171825

Keyword

nonn,tabl

Author

Roger L. Bagula, Dec 19 2009

Extensions

Edited by G. C. Greubel, Feb 22 2021

Status

approved