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

Author

Roger L. Bagula, Dec 19 2009

Keywords

• nonn
• tabl

Examples

			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;
		

Links

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

Crossrefs

Cf. A054142, A085478, A183160.

Programs

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

Mathematica

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

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

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

Extensions

Edited by G. C. Greubel, Feb 22 2021