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%I #8 May 08 2021 01:41:15
%S 1,1,1,1,-3,1,1,6,6,1,1,-15,30,-15,1,1,120,600,600,120,1,1,-63,2520,
%T -6300,2520,-63,1,1,5040,105840,2116800,2116800,105840,5040,1,1,-255,
%U 428400,-4498200,35985600,-4498200,428400,-255,1,1,362880,30844800,25909632000,108820454400,108820454400,25909632000,30844800,362880,1
%N Triangle T(n, k, q) = c(n,q)/( c(k,q)*c(n-k,q) ), where c(n, q) = Product_{j=1..n} f(n, q), f(n, q) = ( (1-q^n)*(1+(-1)^n) + n!*(1-(-1)^n) )/2, and q = 2, read by rows.
%H G. C. Greubel, <a href="/A172427/b172427.txt">Rows n = 0..30 of the triangle, flattened</a>
%F T(n, k, q) = c(n,q)/( c(k,q)*c(n-k,q) ), where c(n, q) = Product_{j=1..n} f(n, q), f(n, q) = ( (1-q^n)*(1+(-1)^n) + n!*(1-(-1)^n) )/2, and q = 2.
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, -3, 1;
%e 1, 6, 6, 1;
%e 1, -15, 30, -15, 1;
%e 1, 120, 600, 600, 120, 1;
%e 1, -63, 2520, -6300, 2520, -63, 1;
%e 1, 5040, 105840, 2116800, 2116800, 105840, 5040, 1;
%e 1, -255, 428400, -4498200, 35985600, -4498200, 428400, -255, 1;
%t f[n_, q_]:= ((1-q^n)*(1+(-1)^n) + n!*(1-(-1)^n))/2;
%t c[n_, q_]:= Product[f[j, q], {j, n}];
%t T[n_, k_, q_]:= c[n, q]/(c[k, q]*c[n-k, q]);
%t Table[T[n, k, 2], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, May 07 2021 *)
%o (Sage)
%o @CachedFunction
%o def f(n,q): return ((1-q^n)*(1+(-1)^n) + factorial(n)*(1-(-1)^n))/2
%o def c(n,q): return product( f(j,q) for j in (1..n) )
%o def T(n,k,q): return c(n,q)/(c(k,q)*c(n-k,q))
%o flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 07 2021
%Y Cf. this sequence (q=3), A172428 (q=4), A172429 (q=5).
%K sign,tabl
%O 0,5
%A _Roger L. Bagula_, Feb 02 2010
%E Edited by _G. C. Greubel_, May 07 2021
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