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 A174449 #13 Sep 08 2022 08:45:51
%S A174449 1,1,1,1,6,1,1,12,12,1,1,20,240,20,1,1,30,600,600,30,1,1,42,1260,
%T A174449 25200,1260,42,1,1,56,2352,70560,70560,2352,56,1,1,72,4032,169344,
%U A174449 5080320,169344,4032,72,1,1,90,6480,362880,15240960,15240960,362880,6480,90,1
%N A174449 Triangle T(n, k, q) = n!*(n+1)!*q^k/((n-k)!(n-k+1)!) if floor(n/2) > k-1, otherwise n!*(n+1)!*q^(n-k)/(k!*(k+1)!) for q = 1, read by rows.
%H A174449 G. C. Greubel, <a href="/A174449/b174449.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A174449 T(n, k, q) = n!*(n+1)!*q^k/((n-k)!(n-k+1)!) if floor(n/2) > k-1, otherwise n!*(n+1)!*q^(n-k)/(k!*(k+1)!) for q = 1.
%F A174449 T(n, n-k, q) = T(n, k, q).
%F A174449 From _G. C. Greubel_, Nov 29 2021: (Start)
%F A174449 T(2*n, n, 1) = A052510(n+1).
%F A174449 T(2*n, n, q) = q^n*(2*n+1)!*Catalan(n) for q = 1.
%F A174449 T(n, k, q) = binomial(n, k)*binomial(n+1, k+1) * ( k!*(k+1)!*q^k/(n-k+1) if (floor(n/2) >= k), otherwise ((n-k)!)^2*q^(n-k) ), for q = 1. (End)
%e A174449 Triangle begins as:
%e A174449 1;
%e A174449 1, 1;
%e A174449 1, 6, 1;
%e A174449 1, 12, 12, 1;
%e A174449 1, 20, 240, 20, 1;
%e A174449 1, 30, 600, 600, 30, 1;
%e A174449 1, 42, 1260, 25200, 1260, 42, 1;
%e A174449 1, 56, 2352, 70560, 70560, 2352, 56, 1;
%e A174449 1, 72, 4032, 169344, 5080320, 169344, 4032, 72, 1;
%e A174449 1, 90, 6480, 362880, 15240960, 15240960, 362880, 6480, 90, 1;
%e A174449 1, 110, 9900, 712800, 39916800, 1676505600, 39916800, 712800, 9900, 110, 1;
%t A174449 T[n_, k_, q_]:= If[Floor[n/2]>k-1, n!*(n+1)!*q^k/((n-k)!*(n-k+1)!), n!*(n+1)!*q^(n-k)/(k!*(k+1)!)];
%t A174449 Table[T[n, k, 1], {n,0,12}, {k,0,n}]//Flatten
%o A174449 (Magma)
%o A174449 F:= Factorial; // T = A174449
%o A174449 T:= func< n,k,q | Floor(n/2) gt k-1 select F(n)*F(n+1)*q^k/(F(n-k)*F(n-k+1)) else F(n)*F(n+1)*q^(n-k)/(F(k)*F(k+1)) >;
%o A174449 [T(n,k,1): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 29 2021
%o A174449 (Sage)
%o A174449 f=factorial
%o A174449 def A174449(n,k,q):
%o A174449 if ((n//2)>k-1): return f(n)*f(n+1)*q^k/(f(n-k)*f(n-k+1))
%o A174449 else: return f(n)*f(n+1)*q^(n-k)/(f(k)*f(k+1))
%o A174449 flatten([[A174449(n,k,1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Nov 29 2021
%Y A174449 Cf. this sequence (q=1), A174450 (q=2), A174451 (q=3).
%Y A174449 Cf. A052510.
%K A174449 nonn,tabl,easy
%O A174449 0,5
%A A174449 _Roger L. Bagula_, Mar 20 2010
%E A174449 Edited by _G. C. Greubel_, Nov 29 2021