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A157640 Triangle of the elementwise product of binomial coefficients with q-binomial coefficients [n,k] for q = 3.

%I A157640 #14 Sep 08 2022 08:45:42
%S A157640 1,1,1,1,8,1,1,39,39,1,1,160,780,160,1,1,605,12100,12100,605,1,1,2184,
%T A157640 165165,677600,165165,2184,1,1,7651,2088723,32401985,32401985,2088723,
%U A157640 7651,1,1,26240,25095280,1405335680,5313925540,1405335680,25095280
%N A157640 Triangle of the elementwise product of binomial coefficients with q-binomial coefficients [n,k] for q = 3.
%C A157640 Row sums are: {1, 2, 10, 80, 1102, 25412, 1012300, 68996720, 8174839942, 1670428649564, 594362629986268,...}.
%C A157640 Other triangles in the family (see name) include: q = 2 (see A157638), q = 3 (this triangle), and q = 4 (see A157641). - _Werner Schulte_, Nov 16 2018
%H A157640 Andrew Howroyd, <a href="/A157640/b157640.txt">Table of n, a(n) for n = 0..1274</a> (rows n=0..49)
%F A157640 T(n,k) = t(n)/(t(k)*t(n-k)) where t(n) = Product_{k=1..n} Sum_{i=0..k-1} k*3^i.
%F A157640 T(n,k) = binomial(n,k) * A022167(n,k) for 0 <= k <= n. - _Werner Schulte_, Nov 16 2018
%e A157640 Triangle begins:
%e A157640   1;
%e A157640   1, 1;
%e A157640   1, 8, 1;
%e A157640   1, 39, 39, 1;
%e A157640   1, 160, 780, 160, 1;
%e A157640   1, 605, 12100, 12100, 605, 1;
%e A157640   1, 2184, 165165, 677600, 165165, 2184, 1;
%e A157640   1, 7651, 2088723, 32401985, 32401985, 2088723, 7651, 1;
%e A157640   1, 26240, 25095280, 1405335680, 5313925540, 1405335680, 25095280, 26240, 1;
%e A157640   ...
%t A157640 t[n_, m_] = Product[Sum[k*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}];
%t A157640 b[n_, k_, m_] = t[n, m]/(t[k, m]*t[n - k, m]);
%t A157640 Flatten[Table[Table[b[n, k, 2], {k, 0, n}], {n, 0, 10}]]
%o A157640 (PARI) T(n, k) = {binomial(n, k)*polcoef(x^k/prod(j=0, k, 1-3^j*x+x*O(x^n)), n)} \\ _Andrew Howroyd_, Nov 19 2018
%o A157640 (PARI) my(q=3); for(n=0,10, for(k=0,n, print1(binomial(n,k)*prod(j=0,k-1, (1-q^(n-j))/(1-q^(j+1))), ", ")); print) \\ _G. C. Greubel_, Nov 17 2018
%o A157640 (Magma) q:=3; [[k le 0 select 1 else Binomial(n,k)*(&*[(1-q^(n-j))/(1-q^(j+1)): j in [0..(k-1)]]): k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, Nov 17 2018
%o A157640 (Sage) [[ binomial(n,k)*gaussian_binomial(n,k).subs(q=3) for k in range(n+1)] for n in range(10)] # _G. C. Greubel_, Nov 17 2018
%Y A157640 Cf. A007318, A022167, A157638, A157641.
%K A157640 nonn,tabl
%O A157640 0,5
%A A157640 _Roger L. Bagula_, Mar 03 2009
%E A157640 Edited and simpler name by _Werner Schulte_ and _Andrew Howroyd_, Nov 19 2018