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A123163
Triangle T(n, k) = binomial((n-k)^2, k^2) read by rows.
2
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
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
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Oct 02 2006
EXTENSIONS
Edited by N. J. A. Sloane, Oct 04 2006
STATUS
approved