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A155516 Triangle T(n, k) = binomial(2*n, 2*k)*binomial(2*n+1, 2*k+1)/(2*n-2*k+1), read by rows.

%I A155516 #10 Sep 08 2022 08:45:40
%S A155516 1,1,1,1,20,1,1,105,105,1,1,336,1764,336,1,1,825,13860,13860,825,1,1,
%T A155516 1716,70785,226512,70785,1716,1,1,3185,273273,2147145,2147145,273273,
%U A155516 3185,1,1,5440,866320,14158144,34763300,14158144,866320,5440,1,1,8721,2372112,71954064,367479684,367479684,71954064,2372112,8721,1
%N A155516 Triangle T(n, k) = binomial(2*n, 2*k)*binomial(2*n+1, 2*k+1)/(2*n-2*k+1), read by rows.
%H A155516 G. C. Greubel, <a href="/A155516/b155516.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A155516 T(n, k) = (2*n + 1)!!/((2*k + 1)!!*(2*(n-k) + 1)!!)*binomial(2*n, 2*k)*binomial(n, k).
%F A155516 From _G. C. Greubel_, May 29 2021: (Start)
%F A155516 T(n, k) = (2*n + 1)!!/((2*k + 1)!!*(2*(n-k) + 1)!!)*A155495(n, k).
%F A155516 T(n, k) = binomial(2*n, 2*k)*binomial(2*n+1, 2*k+1)/(2*n-2*k+1).
%F A155516 Sum_{k=0..n} T(n, k) = Hypergeometric4F3([-n,-n,-n-1/2,-n+1/2], [1/2,1,3/2], 1). (End)
%e A155516 Triangle begins as:
%e A155516   1;
%e A155516   1,    1;
%e A155516   1,   20,       1;
%e A155516   1,  105,     105,        1;
%e A155516   1,  336,    1764,      336,         1;
%e A155516   1,  825,   13860,    13860,       825,         1;
%e A155516   1, 1716,   70785,   226512,     70785,      1716,        1;
%e A155516   1, 3185,  273273,  2147145,   2147145,    273273,     3185,       1;
%e A155516   1, 5440,  866320, 14158144,  34763300,  14158144,   866320,    5440,    1;
%e A155516   1, 8721, 2372112, 71954064, 367479684, 367479684, 71954064, 2372112, 8721, 1;
%t A155516 T[n_, k_]:= Binomial[2*n, 2*k]*Binomial[2*n+1, 2*k+1]/(2*n-2*k+1);
%t A155516 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, May 29 2021 *)
%o A155516 (Magma) [Binomial(2*n, 2*k)*Binomial(2*n+1, 2*k+1)/(2*n-2*k+1): k in [0..n], n in [0..12]]; // _G. C. Greubel_, May 29 2021
%o A155516 (Sage) flatten([[binomial(2*n, 2*k)*binomial(2*n+1, 2*k+1)/(2*n-2*k+1) for k in (0..12)] for n in (0..12)]) # _G. C. Greubel_, May 29 2021
%Y A155516 Cf. A155495.
%K A155516 nonn,tabl,easy
%O A155516 0,5
%A A155516 _Roger L. Bagula_, Jan 23 2009
%E A155516 Edited by _G. C. Greubel_, May 29 2021