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A209330 Triangle defined by T(n,k) = binomial(n^2, n*k), for n>=0, k=0..n, as read by rows.

%I A209330 #22 Oct 16 2018 15:53:43
%S A209330 1,1,1,1,6,1,1,84,84,1,1,1820,12870,1820,1,1,53130,3268760,3268760,
%T A209330 53130,1,1,1947792,1251677700,9075135300,1251677700,1947792,1,1,
%U A209330 85900584,675248872536,39049918716424,39049918716424,675248872536,85900584,1,1
%N A209330 Triangle defined by T(n,k) = binomial(n^2, n*k), for n>=0, k=0..n, as read by rows.
%C A209330 Column 1 equals A014062.
%C A209330 Row sums equal A167009.
%C A209330 Antidiagonal sums equal A209331.
%C A209330 Ignoring initial row T(0,0), equals the logarithmic derivative of the g.f. of triangle A209196.
%H A209330 Paul D. Hanna, <a href="/A209330/b209330.txt">Rows n = 0..30, as a flattened table of n, a(n) for n = 0..495</a>
%e A209330 The triangle of coefficients C(n^2,n*k), n>=k, k=0..n, begins:
%e A209330 1;
%e A209330 1, 1;
%e A209330 1, 6, 1;
%e A209330 1, 84, 84, 1;
%e A209330 1, 1820, 12870, 1820, 1;
%e A209330 1, 53130, 3268760, 3268760, 53130, 1;
%e A209330 1, 1947792, 1251677700, 9075135300, 1251677700, 1947792, 1;
%e A209330 1, 85900584, 675248872536, 39049918716424, 39049918716424, 675248872536, 85900584, 1; ...
%t A209330 Table[Binomial[n^2, n*k], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jan 05 2018 *)
%o A209330 (PARI) {T(n,k)=binomial(n^2,n*k)}
%o A209330 for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
%Y A209330 Cf. A014062 (column 1), A167009 (row sums), A209331, A209196.
%Y A209330 Cf. related triangles: A209196 (exp), A228836, A228832, A226234.
%Y A209330 Cf. A206830.
%K A209330 nonn,tabl
%O A209330 0,5
%A A209330 _Paul D. Hanna_, Mar 06 2012