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%I #13 Jul 20 2021 03:27:58
%S 1,1,1,1,1,1,4,4,4,4,2,1,1,1,2,16,16,8,8,16,16,16,4,16,1,16,4,16,64,
%T 64,64,64,64,64,64,64,16,8,8,8,1,8,8,8,16,256,256,64,64,128,128,64,64,
%U 256,256,256,32,256,16,128,1,128,16,256,32,256
%N Triangle read by rows, 0 <= k <= n: T(n,k) is the denominator of the derivative of the k-th Bernstein basis polynomial of degree n evaluated at the interval midpoint t = 1/2; numerator is A320085.
%H G. C. Greubel, <a href="/A320086/b320086.txt">Rows n = 0..50 of the triangle, flattened</a>
%H Rita T. Farouki, <a href="https://doi.org/10.1016/j.cagd.2012.03.001">The Bernstein polynomial basis: A centennial retrospective</a>, Computer Aided Geometric Design Vol. 29 (2012), 379-419.
%H Ron Goldman, <a href="https://doi.org/10.1016/B978-1-55860-354-7.X5000-4">Pyramid Algorithms. A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling</a>, Morgan Kaufmann Publishers, 2002, Chap. 5.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BernsteinPolynomial.html">Bernstein Polynomial</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Bernstein_polynomial">Bernstein polynomial</a>
%F T(n, k) = denominator of 2*A141692(n,k)/A000079(n).
%F T(n, k) = 2^(n-1)/gcd(n*(binomial(n-1, k-1) - binomial(n-1, k)), 2^(n-1)).
%F T(n, n-k) = T(n,k).
%F T(n, 0) = A084623(n), n > 0.
%F T(2*n+1, 1) = A000302(n).
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, 1, 1;
%e 4, 4, 4, 4;
%e 2, 1, 1, 1, 2;
%e 16, 16, 8, 8, 16, 16;
%e 16, 4, 16, 1, 16, 4, 16;
%e 64, 64, 64, 64, 64, 64, 64, 64;
%e 16, 8, 8, 8, 1, 8, 8, 8, 16;
%e 256, 256, 64, 64, 128, 128, 64, 64, 256, 256;
%e 256, 32, 256, 16, 128, 1, 128, 16, 256, 32, 256;
%e ...
%p T:=proc(n,k) 2^(n-1)/gcd(n*(binomial(n-1,k-1)-binomial(n-1,k)),2^(n-1)); end proc: seq(seq(T(n,k),k=0..n),n=1..11); # _Muniru A Asiru_, Oct 06 2018
%t Table[Denominator[n*(Binomial[n-1, k-1] - Binomial[n-1, k])/2^(n-1)], {n, 0, 12}, {k, 0, n}]//Flatten
%o (Maxima)
%o T(n, k) := 2^(n - 1)/gcd(n*(binomial(n - 1, k - 1) - binomial(n - 1, k)), 2^(n - 1))$
%o tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, n))$
%o (Sage)
%o def A320086(n,k): return denominator(n*(binomial(n-1, k-1) - binomial(n-1, k))/2^(n-1))
%o flatten([[A320086(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jul 19 2021
%Y Inspired by A141692.
%Y Cf. A007318, A128433, A128434, A319861, A319862, A320085.
%K nonn,tabl,easy,frac
%O 0,7
%A _Franck Maminirina Ramaharo_, Oct 05 2018
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