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

%I A123163 #23 Jul 19 2023 08:56:20
%S A123163 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,
%T A123163 12650,11440,0,0,0,0,1,49,58905,2042975,1,0,0,0,0,1,64,211876,
%U A123163 94143280,2042975,0,0,0,0,0,1,81,635376,2054455634,7307872110,1,0,0,0,0,0
%N A123163 Triangle T(n, k) = binomial((n-k)^2, k^2) read by rows.
%H A123163 G. C. Greubel, <a href="/A123163/b123163.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A123163 T(n, k) = (n^2 - 2*n*k + k^2)!/((k^2)!(n^2 - 2*n*k)!).
%F A123163 From _G. C. Greubel_, Jul 18 2023: (Start)
%F A123163 T(n, 0) = T(2*n, n) = 1.
%F A123163 T(n, n) = A000007(n).
%F A123163 Sum_{k=0..n} T(n, k) = A123165(n). (End)
%e A123163 n\k | 0    1      2      3    4    5    6    7
%e A123163 ----+--------------------------------------------
%e A123163   0 | 1;
%e A123163   1 | 1,   0;
%e A123163   2 | 1,   1,     0;
%e A123163   3 | 1,   4,     0,     0;
%e A123163   4 | 1,   9,     1,     0,   0;
%e A123163   5 | 1,  16,   126,     0,   0,   0;
%e A123163   6 | 1,  25,  1820,     1,   0,   0,   0;
%e A123163   7 | 1,  36, 12650, 11440,   0,   0,   0,   0;
%t A123163 T[n_, k_]= (n^2-2*n*k+k^2)!/((k^2)!(n^2-2*n*k)!);
%t A123163 Table[T[n,k], {n,0,10}, {k,0,n}]//Flatten
%t A123163 Flatten[Table[Binomial[(n-m)^2,m^2],{n,0,10},{m,0,n}]] (* _Harvey P. Dale_, Aug 08 2012 *)
%o A123163 (Magma) [Binomial((n-k)^2, k^2): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 18 2023
%o A123163 (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
%Y A123163 Cf. A000007, A011973, A123165.
%K A123163 nonn,tabl
%O A123163 0,8
%A A123163 _Roger L. Bagula_, Oct 02 2006
%E A123163 Edited by _N. J. A. Sloane_, Oct 04 2006