This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A174696 #13 Feb 15 2021 15:38:48
%S A174696 1,1,1,1,49,1,1,841,841,1,1,11881,47881,11881,1,1,161281,1799281,
%T A174696 1799281,161281,1,1,2217601,55560961,154344961,55560961,2217601,1,1,
%U A174696 31570561,1548892801,9680791681,9680791681,1548892801,31570561,1,1,469929601,40967337601,501853968001,1129171881601,501853968001,40967337601,469929601,1
%N A174696 Triangle T(n, k) = n!*(1/k)^2*(binomial(n-1, k-1)*binomial(n, k-1))^2 - n! + 1, read by rows.
%H A174696 G. C. Greubel, <a href="/A174696/b174696.txt">Rows n = 1..100 of the triangle, flattened</a>
%F A174696 T(n, k) = n!*(1/k)^2*(binomial(n-1, k-1)*binomial(n, k-1))^2 - n! + 1.
%F A174696 From _G. C. Greubel_, Feb 09 2021: (Start)
%F A174696 T(n, k) = n! * A174158(n, k) - n! + 1.
%F A174696 Sum_{k=1..n} T(n,k) = n! * Hypergeometric4F3([-n, -n, 1-n, 1-n], [1, 2, 2], 1) - n*(n! - 1) = n! * A319743(n) - n*(n! - 1). (End)
%e A174696 Triangle begins as:
%e A174696 1;
%e A174696 1, 1;
%e A174696 1, 49, 1;
%e A174696 1, 841, 841, 1;
%e A174696 1, 11881, 47881, 11881, 1;
%e A174696 1, 161281, 1799281, 1799281, 161281, 1;
%e A174696 1, 2217601, 55560961, 154344961, 55560961, 2217601, 1;
%e A174696 1, 31570561, 1548892801, 9680791681, 9680791681, 1548892801, 31570561, 1;
%t A174696 T[n_, m_]:= n!*(1/k)^2*(Binomial[n-1, k-1]*Binomial[n, k-1])^2 - n! + 1;
%t A174696 Table[T[n, k], {n,12}, {k,n}]//Flatten
%o A174696 (Sage)
%o A174696 def A174696(n, k): return (factorial(n)/k^2)*(binomial(n-1, k-1)*binomial(n, k-1))^2 - factorial(n) + 1
%o A174696 flatten([[A174696(n, k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Feb 09 2021
%o A174696 (Magma)
%o A174696 A174696:= func< n, k | (Factorial(n)/k^2)*(Binomial(n-1, k-1)*Binomial(n, k-1))^2 - Factorial(n) + 1 >;
%o A174696 [A174696(n, k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Feb 09 2021
%Y A174696 Cf. A174158, A174694, A176013, A319743.
%K A174696 nonn,tabl,easy
%O A174696 1,5
%A A174696 _Roger L. Bagula_, Mar 27 2010
%E A174696 Edited by _G. C. Greubel_, Feb 09 2021