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A228836 Triangle defined by T(n,k) = binomial(n^2, (n-k)*k), for n>=0, k=0..n, as read by rows.

%I A228836 #15 Aug 02 2025 10:01:44
%S A228836 1,1,1,1,4,1,1,36,36,1,1,560,1820,560,1,1,12650,177100,177100,12650,1,
%T A228836 1,376992,30260340,94143280,30260340,376992,1,1,13983816,8217822536,
%U A228836 92263734836,92263734836,8217822536,13983816,1,1,621216192,3284214703056,159518999862720,488526937079580,159518999862720,3284214703056,621216192,1
%N A228836 Triangle defined by T(n,k) = binomial(n^2, (n-k)*k), for n>=0, k=0..n, as read by rows.
%H A228836 Paul D. Hanna, <a href="/A228836/b228836.txt">Rows 0..30 as a flattened table of n, a(n) for n = 0..495</a>
%e A228836 The triangle of coefficients C(n^2, (n-k)*k), n>=k, k=0..n, begins:
%e A228836   1;
%e A228836   1, 1;
%e A228836   1, 4, 1;
%e A228836   1, 36, 36, 1;
%e A228836   1, 560, 1820, 560, 1;
%e A228836   1, 12650, 177100, 177100, 12650, 1;
%e A228836   1, 376992, 30260340, 94143280, 30260340, 376992, 1;
%e A228836   1, 13983816, 8217822536, 92263734836, 92263734836, 8217822536, 13983816, 1;
%e A228836   ...
%t A228836 T[n_,k_]:=Binomial[n^2, (n-k)*k]; Table[T[n,k],{n,0,8},{k,0,n}]//Flatten (* _Stefano Spezia_, Aug 02 2025 *)
%o A228836 (PARI) {T(n,k)=binomial(n^2, (n-k)*k)}
%o A228836 for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
%Y A228836 Cf. A207136 (row sums), A228837 (antidiagonal sums), A070780 (column 1).
%Y A228836 Cf. related triangles: A228900(exp), A209330, A226234, A228832.
%K A228836 nonn,tabl
%O A228836 0,5
%A A228836 _Paul D. Hanna_, Sep 05 2013