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%I #9 Sep 08 2022 08:45:51
%S 1,1,1,1,18,1,1,36,36,1,1,60,2160,60,1,1,90,5400,5400,90,1,1,126,
%T 11340,680400,11340,126,1,1,168,21168,1905120,1905120,21168,168,1,1,
%U 216,36288,4572288,411505920,4572288,36288,216,1,1,270,58320,9797760,1234517760,1234517760,9797760,58320,270,1
%N 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 = 3, read by rows.
%H G. C. Greubel, <a href="/A174451/b174451.txt">Rows n = 0..50 of the triangle, flattened</a>
%F 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 = 3.
%F T(n, n-k, q) = T(n, k, q).
%F From _G. C. Greubel_, Nov 29 2021: (Start)
%F T(2*n, n, q) = q^n*(2*n+1)!*Catalan(n) for q = 3.
%F 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 = 3. (End)
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 18, 1;
%e 1, 36, 36, 1;
%e 1, 60, 2160, 60, 1;
%e 1, 90, 5400, 5400, 90, 1;
%e 1, 126, 11340, 680400, 11340, 126, 1;
%e 1, 168, 21168, 1905120, 1905120, 21168, 168, 1;
%e 1, 216, 36288, 4572288, 411505920, 4572288, 36288, 216, 1;
%e 1, 270, 58320, 9797760, 1234517760, 1234517760, 9797760, 58320, 270, 1;
%t 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 Table[T[n, k, 3], {n,0,12}, {k,0,n}]//Flatten
%o (Magma)
%o F:= Factorial; // T = A174451
%o 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 [T(n,k,3): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 29 2021
%o (Sage)
%o f=factorial
%o def A174451(n,k,q):
%o if ((n//2)>k-1): return f(n)*f(n+1)*q^k/(f(n-k)*f(n-k+1))
%o else: return f(n)*f(n+1)*q^(n-k)/(f(k)*f(k+1))
%o flatten([[A174451(n,k,3) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Nov 29 2021
%Y Cf. A174449 (q=1), A174450 (q=2), this sequence (q=3).
%Y Cf. A000108.
%K nonn,tabl,easy
%O 0,5
%A _Roger L. Bagula_, Mar 20 2010
%E Edited by _G. C. Greubel_, Nov 29 2021
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