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

0,8

Links

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

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)

Example

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;

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 *)

Prog

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

Crossrefs

Cf. A000007, A011973, A123165.

Sequence in context: A344386 A046783 A134832 * A194794 A337967 A317448

Adjacent sequences: A123160 A123161 A123162 * A123164 A123165 A123166

Keyword

nonn,tabl

Author

Roger L. Bagula, Oct 02 2006

Extensions

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

Status

approved