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%I #13 Jul 20 2021 03:27:50
%S 0,-1,1,-1,0,1,-3,-3,3,3,-1,-1,0,1,1,-5,-15,-5,5,15,5,-3,-3,-15,0,15,
%T 3,3,-7,-35,-63,-35,35,63,35,7,-1,-3,-7,-7,0,7,7,3,1,-9,-63,-45,-63,
%U -63,63,63,45,63,9,-5,-5,-135,-15,-105,0,105,15,135,5,5
%N Triangle read by rows, 0 <= k <= n: T(n,k) is the numerator of the derivative of the k-th Bernstein basis polynomial of degree n evaluated at the interval midpoint t = 1/2; denominator is A320086.
%C If n = 2*k, then T(n,k) = 0 since the k-th Bernstein basis polynomial of degree n has a single unique local maximum occurring at t = k/n, which coincides with the interval midpoint t = 1/2 (T(0,0) = 0 because the only 0 degree Bernstein basis polynomial is the constant 1).
%H G. C. Greubel, <a href="/A320085/b320085.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) = numerator of 2*A141692(n,k)/A000079(n).
%F T(n, k) = n*(binomial(n-1, k-1) - binomial(n-1, k))/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) = -n.
%F T(2*n+1, 1) = -A000466(n).
%F T(2*n, 1) = -A069834(n-1), n > 1.
%F T(n, k)/A320086(n,k) = 4*n*(k/n - 1/2)*A319861(n,k)/A319861(n,k).
%F Sum_{k=0..n} k*T(n,k)/A320086(n,k) = n.
%F Sum_{k=0..n} k^2*T(n,k)/A320086(n,k) = n^2.
%F Sum_{k=0..n} k*(k-1)*T(n,k)/A320086(n,k) = n*(n - 1).
%e Triangle begins:
%e 0;
%e -1, 1;
%e -1, 0, 1;
%e -3, -3, 3, 3;
%e -1, -1, 0, 1, 1;
%e -5, -15, -5, 5, 15, 5;
%e -3, -3, -15, 0, 15, 3, 3;
%e -7, -35, -63, -35, 35, 63, 35, 7;
%e -1, -3, -7, -7, 0, 7, 7, 3, 1;
%e -9, -63, -45, -63, -63, 63, 63, 45, 63, 9;
%e -5, -5, -135, -15, -105, 0, 105, 15, 135, 5, 5;
%e ...
%p T:=proc(n,k) n*(binomial(n-1,k-1)-binomial(n-1,k))/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=0..11); # _Muniru A Asiru_, Oct 06 2018
%t Table[Numerator[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) := n*(binomial(n - 1, k - 1) - binomial(n - 1, k))/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 A320085(n,k): return numerator(n*(binomial(n-1, k-1) - binomial(n-1, k))/2^(n-1))
%o flatten([[A320085(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, A320086.
%K sign,easy,tabl,frac
%O 0,7
%A _Franck Maminirina Ramaharo_, Oct 05 2018
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