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A174376 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.

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