A123163 Triangle T(n, k) = binomial((n-k)^2, k^2) read by rows.

1, 1, 0, 1, 1, 0, 1, 4, 0, 0, 1, 9, 1, 0, 0, 1, 16, 126, 0, 0, 0, 1, 25, 1820, 1, 0, 0, 0, 1, 36, 12650, 11440, 0, 0, 0, 0, 1, 49, 58905, 2042975, 1, 0, 0, 0, 0, 1, 64, 211876, 94143280, 2042975, 0, 0, 0, 0, 0, 1, 81, 635376, 2054455634, 7307872110, 1, 0, 0, 0, 0, 0

Offset: 0

Author

Roger L. Bagula, Oct 02 2006

Keywords

• nonn
• tabl

Examples

			n\k | 0    1      2      3    4    5    6    7
----+--------------------------------------------
  0 | 1;
  1 | 1,   0;
  2 | 1,   1,     0;
  3 | 1,   4,     0,     0;
  4 | 1,   9,     1,     0,   0;
  5 | 1,  16,   126,     0,   0,   0;
  6 | 1,  25,  1820,     1,   0,   0,   0;
  7 | 1,  36, 12650, 11440,   0,   0,   0,   0;
		

Links

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

Crossrefs

Cf. A000007, A011973, A123165.

Programs

Magma

[Binomial((n-k)^2, k^2): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 18 2023

Mathematica

T[n_, k_]= (n^2-2*n*k+k^2)!/((k^2)!(n^2-2*n*k)!);
Table[T[n,k], {n,0,10}, {k,0,n}]//Flatten
Flatten[Table[Binomial[(n-m)^2,m^2],{n,0,10},{m,0,n}]] (* Harvey P. Dale, Aug 08 2012 *)

SageMath

flatten([[binomial((n-k)^2, k^2) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 18 2023

Formula

T(n, k) = (n^2 - 2*n*k + k^2)!/((k^2)!(n^2 - 2*n*k)!).

From G. C. Greubel, Jul 18 2023: (Start)

T(n, 0) = T(2*n, n) = 1.

T(n, n) = A000007(n).

Sum_{k=0..n} T(n, k) = A123165(n). (End)

Extensions

Edited by N. J. A. Sloane, Oct 04 2006