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%I #16 Sep 08 2022 08:45:42
%S 1,1,1,1,10,1,1,63,63,1,1,340,2142,340,1,1,1705,57970,57970,1705,1,1,
%T 8190,1396395,7536100,1396395,8190,1,1,38227,31307913,847301455,
%U 847301455,31307913,38227,1,1,174760,668055052,86847156760,435512947870
%N Triangle of the elementwise product of binomial coefficients with q-binomial coefficients [n,k] for q = 4.
%C Row sums are: {1, 2, 12, 128, 2824, 119352, 10345272, 1757295192, 610543721016, 418465696229912, 584788183756728952,...}.
%C Other triangles in the family (see name) include: q = 2 (see A157638), q = 3 (see A157640), and q = 4 (this triangle). - Werner Schulte, Nov 16 2018
%H Andrew Howroyd, <a href="/A157641/b157641.txt">Table of n, a(n) for n = 0..1274</a>
%F T(n,k) = t(n)/(t(k)*t(n-k)) where t(n) = Product_{k=1..n} Sum_{i=0..k-1} k*4^i.
%F T(n,k) = binomial(n,k) * A022168(n,k) for 0 <= k <= n. - _Werner Schulte_, Nov 16 2018
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, 10, 1;
%e 1, 63, 63, 1;
%e 1, 340, 2142, 340, 1;
%e 1, 1705, 57970, 57970, 1705, 1;
%e 1, 8190, 1396395, 7536100, 1396395, 8190, 1;
%e 1, 38227, 31307913, 847301455, 847301455, 31307913, 38227, 1;
%e ...
%t t[n_, m_] = Product[Sum[k*(m + 1)^i, {i, 0, k - 1}], {k, 1, n}];
%t b[n_, k_, m_] = t[n, m]/(t[k, m]*t[n - k, m]);
%t Flatten[Table[Table[b[n, k, 3], {k, 0, n}], {n, 0, 10}]]
%o (PARI) T(n, k) = {binomial(n, k)*polcoef(x^k/prod(j=0, k, 1-4^j*x+x*O(x^n)), n)} \\ _Andrew Howroyd_, Nov 19 2018
%o (PARI) my(q=4); 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 (Magma) q:=4; [[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 (Sage) [[ binomial(n,k)*gaussian_binomial(n,k).subs(q=4) for k in range(n+1)] for n in range(10)] # _G. C. Greubel_, Nov 17 2018
%Y Cf. A007318, A022168, A157638, A157640.
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, Mar 03 2009
%E Edited and simpler name by _Werner Schulte_ and _Andrew Howroyd_, Nov 19 2018
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