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A157114 Triangle T(n, k) = binomial(n*k, n-k) + binomial(n*(n-k), k), read by rows.

%I A157114 #6 Mar 09 2021 03:35:56
%S A157114 2,1,1,1,4,1,1,9,9,1,1,16,56,16,1,1,25,225,225,25,1,1,36,771,1632,771,
%T A157114 36,1,1,49,2597,9261,9261,2597,49,1,1,64,9136,52384,71920,52384,9136,
%U A157114 64,1,1,81,33777,320814,525987,525987,320814,33777,81,1,1,100,129130,2090540,4326015,4237520,4326015,2090540,129130,100,1
%N A157114 Triangle T(n, k) = binomial(n*k, n-k) + binomial(n*(n-k), k), read by rows.
%H A157114 G. C. Greubel, <a href="/A157114/b157114.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A157114 T(n, k) = binomial(n*k, n-k) + binomial(n*(n-k), k).
%F A157114 Sum_{k=0..n} T(n,k) = 2*A099237(n). - _G. C. Greubel_, Mar 09 2021
%e A157114 Triangle begins as:
%e A157114   2;
%e A157114   1,   1;
%e A157114   1,   4,      1;
%e A157114   1,   9,      9,       1;
%e A157114   1,  16,     56,      16,       1;
%e A157114   1,  25,    225,     225,      25,       1;
%e A157114   1,  36,    771,    1632,     771,      36,       1;
%e A157114   1,  49,   2597,    9261,    9261,    2597,      49,       1;
%e A157114   1,  64,   9136,   52384,   71920,   52384,    9136,      64,      1;
%e A157114   1,  81,  33777,  320814,  525987,  525987,  320814,   33777,     81,   1;
%e A157114   1, 100, 129130, 2090540, 4326015, 4237520, 4326015, 2090540, 129130, 100, 1;
%p A157114 A157114:= (n,k) -> binomial(n*k, n-k) + binomial(n*(n-k), k);
%p A157114 seq(seq(A157114(n,k), k=0..n), n=0..12); # _G. C. Greubel_, Mar 09 2021
%t A157114 T[n_, k_]:= Binomial[n*k, n-k], Binomial[n*(n-k), k];
%t A157114 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Mar 09 2021 *)
%o A157114 (Sage)
%o A157114 def A157114(n,k): return binomial(n*k, n-k) + binomial(n*(n-k), k)
%o A157114 flatten([[A157114(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 09 2021
%o A157114 (Magma)
%o A157114 A157114:= func< n,k | Binomial(n*k, n-k) + Binomial(n*(n-k), k) >;
%o A157114 [A157114(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 09 2021
%Y A157114 Cf. A099237.
%K A157114 nonn,tabl
%O A157114 0,1
%A A157114 _Roger L. Bagula_, Feb 23 2009
%E A157114 Edited by _G. C. Greubel_, Mar 09 2021