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%I #6 Nov 28 2021 03:29:55
%S 1,1,1,1,4,1,1,6,6,1,1,8,48,8,1,1,10,80,80,10,1,1,12,120,960,120,12,1,
%T 1,14,168,1680,1680,168,14,1,1,16,224,2688,26880,2688,224,16,1,1,18,
%U 288,4032,48384,48384,4032,288,18,1,1,20,360,5760,80640,967680,80640,5760,360,20,1
%N Triangle T(n, k) = n!*q^k/(n-k)! if floor(n/2) > k-1 otherwise n!*q^(n-k)/k!, with q = 2, read by rows.
%C Row sums are: {1, 2, 6, 14, 66, 182, 1226, 3726, 32738, 105446, 1141242, ...}.
%H G. C. Greubel, <a href="/A174376/b174376.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n, k) = n!*q^k/(n-k)! if floor(n/2) > k-1 otherwise n!*q^(n-k)/k!, with q = 2.
%F T(n, n-k) = T(n, k).
%F T(2*n, n) = A052714(n+1). - _G. C. Greubel_, Nov 28 2021
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 4, 1;
%e 1, 6, 6, 1;
%e 1, 8, 48, 8, 1;
%e 1, 10, 80, 80, 10, 1;
%e 1, 12, 120, 960, 120, 12, 1;
%e 1, 14, 168, 1680, 1680, 168, 14, 1;
%e 1, 16, 224, 2688, 26880, 2688, 224, 16, 1;
%e 1, 18, 288, 4032, 48384, 48384, 4032, 288, 18, 1;
%e 1, 20, 360, 5760, 80640, 967680, 80640, 5760, 360, 20, 1;
%t T[n_, k_, q_]:= If[Floor[n/2]>=k, n!*q^k/(n-k)!, n!*q^(n-k)/k!];
%t Table[T[n, k, 2], {n,0,12}, {k,0,n}]//Flatten
%o (Sage)
%o f=factorial
%o def T(n,k,q): return f(n)*q^k/f(n-k) if ((n//2)>k-1) else f(n)*q^(n-k)/f(k)
%o flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Nov 28 2021
%Y Cf. A159623 (q=1), this sequence (q=2), A174377 (q=3), A174378 (q=4).
%Y Cf. A052714.
%K nonn,tabl,easy
%O 0,5
%A _Roger L. Bagula_, Mar 17 2010
%E Edited by _G. C. Greubel_, Nov 28 2021
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