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Triangle of the elementwise product of binomial coefficients with q-binomial coefficients [n,k] for q = 2.
3

%I #27 Sep 08 2022 08:45:42

%S 1,1,1,1,6,1,1,21,21,1,1,60,210,60,1,1,155,1550,1550,155,1,1,378,9765,

%T 27900,9765,378,1,1,889,56007,413385,413385,56007,889,1,1,2040,302260,

%U 5440680,14055090,5440680,302260,2040,1,1,4599,1563660,66194940

%N Triangle of the elementwise product of binomial coefficients with q-binomial coefficients [n,k] for q = 2.

%C Other triangles in the family (see name) include: q = 2 (this triangle), q = 3 (see A157640), and q = 4 (see A157641). - _Werner Schulte_, Nov 16 2018

%H Andrew Howroyd, <a href="/A157638/b157638.txt">Rows n=0..49 of triangle, flattened</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*2^i.

%F T(n,k) = binomial(n,k)*A022166(n,k) for 0 <= k <= n. - _Werner Schulte_, Nov 14 2018

%F T(n,k) = n!*A005329(n)/(k!*A005329(k)*(n-k)!*A005329(n-k)). - _Andrew Howroyd_, Nov 14 2018

%e Triangle begins:

%e 1;

%e 1, 1;

%e 1, 6, 1;

%e 1, 21, 21, 1;

%e 1, 60, 210, 60, 1;

%e 1, 155, 1550, 1550, 155, 1;

%e 1, 378, 9765, 27900, 9765, 378, 1;

%e 1, 889, 56007, 413385, 413385, 56007, 889, 1;

%e 1, 2040, 302260, 5440680, 14055090, 5440680, 302260, 2040, 1;

%e 1, 4599, 1563660, 66194940, 417028122, 417028122, 66194940, 1563660, 4599, 1;

%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, 1], {k, 0, n}], {n, 0, 10}]]

%o (PARI) T(n,k) = {binomial(n,k)*polcoef(x^k/prod(j=0, k, 1-2^j*x+x*O(x^n)), n)} \\ _Andrew Howroyd_, Nov 14 2018

%o (PARI) q=2; 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))), ", "))) \\ _G. C. Greubel_, Nov 17 2018

%o (Magma) q:=2; [[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=2) for k in range(n+1)] for n in range(10)] # _G. C. Greubel_, Nov 17 2018

%Y Cf. A007318, A005329, A022166, A157640, A157641.

%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 14 2018