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 A174689 #8 Feb 10 2021 01:38:49
%S A174689 1,1,1,1,7,1,1,49,49,1,1,361,841,361,1,1,2881,11881,11881,2881,1,1,
%T A174689 25201,161281,287281,161281,25201,1,1,241921,2217601,6168961,6168961,
%U A174689 2217601,241921,1,1,2540161,31570561,126403201,197527681,126403201,31570561,2540161,1
%N A174689 Triangle T(n, k) = n! * binomial(n, k)^2 - n! + 1, read by rows.
%H A174689 G. C. Greubel, <a href="/A174689/b174689.txt">Rows n = 0..100 of the triangle, flattened</a>
%F A174689 T(n, k) = n! * binomial(n, k)^2 - n! + 1.
%F A174689 From _G. C. Greubel_, Feb 10 2021: (Start)
%F A174689 T(n, k) = n! * ( A008459(n, k) - 1 ) + 1.
%F A174689 Sum_{k=0..n} T(n, k) = (n+1)*( n!*( C_{n} - 1 ) + 1 ) = (n+1)*( n!*( A000108(n) - 1 ) + 1). (End)
%e A174689 Triangle begins as:
%e A174689 1;
%e A174689 1, 1;
%e A174689 1, 7, 1;
%e A174689 1, 49, 49, 1;
%e A174689 1, 361, 841, 361, 1;
%e A174689 1, 2881, 11881, 11881, 2881, 1;
%e A174689 1, 25201, 161281, 287281, 161281, 25201, 1;
%e A174689 1, 241921, 2217601, 6168961, 6168961, 2217601, 241921, 1;
%e A174689 1, 2540161, 31570561, 126403201, 197527681, 126403201, 31570561, 2540161, 1;
%t A174689 T[n_, k_]:= n!*Binomial[n, k]^2 - n! + 1;
%t A174689 Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten
%o A174689 (Sage) flatten([[factorial(n)*(binomial(n, k)^2 -1) + 1 for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 10 2021
%o A174689 (Magma) [Factorial(n)*(Binomial(n, k)^2 -1) + 1: k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 10 2021
%Y A174689 Cf. A000108, A008459, A174690.
%K A174689 nonn,tabl,easy
%O A174689 0,5
%A A174689 _Roger L. Bagula_, Mar 27 2010
%E A174689 Edited by _G. C. Greubel_, Feb 10 2021