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%I #6 Jan 09 2022 02:32:03
%S 1,1,1,1,3,1,1,21,21,1,1,315,2205,315,1,1,9765,1025325,1025325,9765,1,
%T 1,615195,2002459725,30036895875,2002459725,615195,1,1,78129765,
%U 16021680259725,7450081320772125,7450081320772125,16021680259725,78129765,1
%N Triangle T(n, k, q) = t(n,q)/(t(k,q)*t(n-k,q)), where t(n, k) = Product_{j=1..n} q-Pochhammer(j, k+1, k+1)/(1-(k+1))^j and t(n, 0) = n!, with q = 1, read by rows.
%D Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), page 182.
%H G. C. Greubel, <a href="/A156950/b156950.txt">Rows n = 0..20 of the triangle, flattened</a>
%F T(n, k, q) = t(n,q)/(t(k,q)*t(n-k,q)), where t(n, k) = Product_{j=1..n} q-Pochhammer(j, k+1, k+1)/(1-(k+1))^j and t(n, 0) = n!, with q = 1.
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 21, 21, 1;
%e 1, 315, 2205, 315, 1;
%e 1, 9765, 1025325, 1025325, 9765, 1;
%e 1, 615195, 2002459725, 30036895875, 2002459725, 615195, 1;
%t t[n_, k_]= If[k==0, n!, Product[QPochhammer[k+1, k+1, j]/(-k)^j, {j, n}]];
%t T[n_, k_, q_]= t[n, q]/(t[k, q]*t[n-k, q]);
%t Table[T[n,k,1], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jan 08 2022 *)
%o (Sage)
%o from sage.combinat.q_analogues import q_pochhammer
%o def t(n,k): return factorial(n) if (k==0) else product( q_pochhammer(j, k+1, k+1)/(-k)^j for j in (1..n) )
%o def T(n,k,q): return t(n,q)/(t(k,q)*t(n-k,q))
%o flatten([[T(n,k,1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jan 08 2022
%Y Cf. A007318 (q=0), this sequence (q=1), A156951 (q=2), A156952 (q=3).
%Y Cf. A156953.
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, Feb 19 2009
%E Edited by _G. C. Greubel_, Jan 08 2022
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