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A174345 Triangle T(n, k) = (1/k)*binomial(n-1, k-1)*binomial(n, k-1)*2^(k-1) if floor(n/2) >= k, otherwise (1/k)*binomial(n-1, k-1)*binomial(n, k-1)*2^(n-k), read by rows.

%I A174345 #7 Nov 28 2021 03:03:12
%S A174345 1,1,1,1,6,1,1,12,12,1,1,20,80,20,1,1,30,200,200,30,1,1,42,420,1400,
%T A174345 420,42,1,1,56,784,3920,3920,784,56,1,1,72,1344,9408,28224,9408,1344,
%U A174345 72,1,1,90,2160,20160,84672,84672,20160,2160,90,1
%N A174345 Triangle T(n, k) = (1/k)*binomial(n-1, k-1)*binomial(n, k-1)*2^(k-1) if floor(n/2) >= k, otherwise (1/k)*binomial(n-1, k-1)*binomial(n, k-1)*2^(n-k), read by rows.
%H A174345 G. C. Greubel, <a href="/A174345/b174345.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A174345 T(n, k) = (1/k)*binomial(n-1, k-1)*binomial(n, k-1)*2^(k-1) if floor(n/2) >= k, otherwise (1/k)*binomial(n-1, k-1)*binomial(n, k-1)*2^(n-k).
%F A174345 T(n, n-k) = T(n, k).
%F A174345 From _G. C. Greubel_, Nov 28 2021: (Start)
%F A174345 T(n, n-1) = A180291(n), n > 1.
%F A174345 T(n, n-1) = 2*A000217(n-1), n > 2. (End)
%e A174345 Triangle begins as:
%e A174345   1;
%e A174345   1,  1;
%e A174345   1,  6,    1;
%e A174345   1, 12,   12,     1;
%e A174345   1, 20,   80,    20,     1;
%e A174345   1, 30,  200,   200,    30,     1;
%e A174345   1, 42,  420,  1400,   420,    42,     1;
%e A174345   1, 56,  784,  3920,  3920,   784,    56,    1;
%e A174345   1, 72, 1344,  9408, 28224,  9408,  1344,   72,  1;
%e A174345   1, 90, 2160, 20160, 84672, 84672, 20160, 2160, 90,  1;
%t A174345 Table[(Binomial[n-1, k-1]*Binomial[n, k-1]/k)*If[Floor[n/2]>=k, 2^(k-1), 2^(n-k)], {n,12}, {k,n}]//Flatten
%o A174345 (Sage)
%o A174345 def A174345(n,k):
%o A174345     b=binomial
%o A174345     if ((n//2)>k-1): return (1/(n+1))*b(n-1, k-1)*b(n+1, k)*2^(k-1)
%o A174345     else: return (1/(n+1))*b(n-1, k-1)*b(n+1, k)*2^(n-k)
%o A174345 flatten([[A174345(n,k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Nov 28 2021
%Y A174345 Cf. A000217, A180291.
%K A174345 nonn,tabl,easy
%O A174345 1,5
%A A174345 _Roger L. Bagula_, Mar 16 2010
%E A174345 Edited by _G. C. Greubel_, Nov 28 2021