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A123163
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Triangle T(n, k) = binomial((n-k)^2, k^2) read by rows.
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2
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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
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OFFSET
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0,8
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LINKS
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FORMULA
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T(n, k) = (n^2 - 2*n*k + k^2)!/((k^2)!(n^2 - 2*n*k)!).
T(n, 0) = T(2*n, n) = 1.
Sum_{k=0..n} T(n, k) = A123165(n). (End)
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EXAMPLE
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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;
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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