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A320086 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.

%I A320086 #14 Feb 16 2025 08:33:56
%S A320086 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 A320086 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 A320086 256,256,256,32,256,16,128,1,128,16,256,32,256
%N A320086 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 A320086 G. C. Greubel, <a href="/A320086/b320086.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A320086 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 A320086 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 A320086 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BernsteinPolynomial.html">Bernstein Polynomial</a>
%H A320086 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bernstein_polynomial">Bernstein polynomial</a>
%F A320086 T(n, k) = denominator of 2*A141692(n,k)/A000079(n).
%F A320086 T(n, k) = 2^(n-1)/gcd(n*(binomial(n-1, k-1) - binomial(n-1, k)), 2^(n-1)).
%F A320086 T(n, n-k) = T(n,k).
%F A320086 T(n, 0) = A084623(n), n > 0.
%F A320086 T(2*n+1, 1) = A000302(n).
%e A320086 Triangle begins:
%e A320086     1;
%e A320086     1,   1;
%e A320086     1,   1,   1;
%e A320086     4,   4,   4,  4;
%e A320086     2,   1,   1,  1,   2;
%e A320086    16,  16,   8,  8,  16,  16;
%e A320086    16,   4,  16,  1,  16,   4,  16;
%e A320086    64,  64,  64, 64,  64,  64,  64, 64;
%e A320086    16,   8,   8,  8,   1,   8,   8,  8,  16;
%e A320086   256, 256,  64, 64, 128, 128,  64, 64, 256, 256;
%e A320086   256,  32, 256, 16, 128,   1, 128, 16, 256,  32, 256;
%e A320086   ...
%p A320086 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 A320086 Table[Denominator[n*(Binomial[n-1, k-1] - Binomial[n-1, k])/2^(n-1)], {n, 0, 12}, {k, 0, n}]//Flatten
%o A320086 (Maxima)
%o A320086 T(n, k) := 2^(n - 1)/gcd(n*(binomial(n - 1, k - 1) - binomial(n - 1, k)), 2^(n - 1))$
%o A320086 tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, n))$
%o A320086 (Sage)
%o A320086 def A320086(n,k): return denominator(n*(binomial(n-1, k-1) - binomial(n-1, k))/2^(n-1))
%o A320086 flatten([[A320086(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jul 19 2021
%Y A320086 Inspired by A141692.
%Y A320086 Cf. A007318, A128433, A128434, A319861, A319862, A320085.
%K A320086 nonn,tabl,easy,frac
%O A320086 0,7
%A A320086 _Franck Maminirina Ramaharo_, Oct 05 2018