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A245243 Triangle, read by rows, defined by T(n,k) = C(n^2 - k^2, n*k - k^2), for k=0..n, n>=0.

%I A245243 #20 Jan 06 2019 11:27:36
%S A245243 1,1,1,1,3,1,1,28,10,1,1,455,495,35,1,1,10626,54264,8008,126,1,1,
%T A245243 324632,10518300,4686825,125970,462,1,1,12271512,3190187286,
%U A245243 5586853480,354817320,1961256,1716,1,1,553270671,1399358844975,11899700525790,2254848913647,25140840660,30421755,6435,1
%N A245243 Triangle, read by rows, defined by T(n,k) = C(n^2 - k^2, n*k - k^2), for k=0..n, n>=0.
%C A245243 Row sums equal A245242.
%C A245243 Central terms are A245245(n) = C(3*n^2, n^2).
%H A245243 Paul D. Hanna, <a href="/A245243/b245243.txt">Table of n, a(n) for rows 0..30 of flattened triangle.</a>
%F A245243 T(n,k) = C(n^2, n*k) * C(n*k, k^2) / C(n^2, k^2).
%F A245243 T(n,k) = (n^2 - k^2)! / ( (n^2 - n*k)! * (n*k - k^2)! ).
%F A245243 T(n,k) = ((n+k)*(n-k))! / ( (n*(n-k))! * (k*(n-k))! ).
%e A245243 Triangle T(n,k) = C(n^2 - k^2, n*k - k^2) begins:
%e A245243 1;
%e A245243 1, 1;
%e A245243 1, 3, 1;
%e A245243 1, 28, 10, 1;
%e A245243 1, 455, 495, 35, 1;
%e A245243 1, 10626, 54264, 8008, 126, 1;
%e A245243 1, 324632, 10518300, 4686825, 125970, 462, 1;
%e A245243 1, 12271512, 3190187286, 5586853480, 354817320, 1961256, 1716, 1;
%e A245243 1, 553270671, 1399358844975, 11899700525790, 2254848913647, 25140840660, 30421755, 6435, 1; ...
%t A245243 Table[Binomial[n^2-k^2,n k-k^2],{n,0,10},{k,0,n}]//Flatten (* _Harvey P. Dale_, Jan 06 2019 *)
%o A245243 (PARI) {T(n,k) = binomial(n^2 - k^2, n*k - k^2)}
%o A245243 for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
%o A245243 (PARI) {T(n,k) = binomial(n^2,n*k) * binomial(n*k,k^2) / binomial(n^2,k^2)}
%o A245243 for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
%Y A245243 Cf. A245242 (row sums), A245245 (central terms), A209330, A228832.
%K A245243 nonn,tabl
%O A245243 0,5
%A A245243 _Paul D. Hanna_, Jul 14 2014