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A320085 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.

%I A320085 #14 Feb 16 2025 08:33:56
%S A320085 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 A320085 3,3,-7,-35,-63,-35,35,63,35,7,-1,-3,-7,-7,0,7,7,3,1,-9,-63,-45,-63,
%U A320085 -63,63,63,45,63,9,-5,-5,-135,-15,-105,0,105,15,135,5,5
%N A320085 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 A320085 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 A320085 G. C. Greubel, <a href="/A320085/b320085.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A320085 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 A320085 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 A320085 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BernsteinPolynomial.html">Bernstein Polynomial</a>
%H A320085 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bernstein_polynomial">Bernstein polynomial</a>
%F A320085 T(n, k) = numerator of 2*A141692(n,k)/A000079(n).
%F A320085 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 A320085 T(n, n-k) = -T(n,k).
%F A320085 T(n, 0) = -n.
%F A320085 T(2*n+1, 1) = -A000466(n).
%F A320085 T(2*n, 1) = -A069834(n-1), n > 1.
%F A320085 T(n, k)/A320086(n,k) = 4*n*(k/n - 1/2)*A319861(n,k)/A319861(n,k).
%F A320085 Sum_{k=0..n} k*T(n,k)/A320086(n,k) = n.
%F A320085 Sum_{k=0..n} k^2*T(n,k)/A320086(n,k) = n^2.
%F A320085 Sum_{k=0..n} k*(k-1)*T(n,k)/A320086(n,k) = n*(n - 1).
%e A320085 Triangle begins:
%e A320085    0;
%e A320085   -1,   1;
%e A320085   -1,   0,    1;
%e A320085   -3,  -3,    3,   3;
%e A320085   -1,  -1,    0,   1,    1;
%e A320085   -5, -15,   -5,   5,   15,  5;
%e A320085   -3,  -3,  -15,   0,   15,  3,   3;
%e A320085   -7, -35,  -63, -35,   35, 63,  35,  7;
%e A320085   -1,  -3,   -7,  -7,    0,  7,   7,  3,   1;
%e A320085   -9, -63,  -45, -63,  -63, 63,  63, 45,  63, 9;
%e A320085   -5,  -5, -135, -15, -105,  0, 105, 15, 135, 5, 5;
%e A320085   ...
%p A320085 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 A320085 Table[Numerator[n*(Binomial[n-1, k-1] - Binomial[n-1, k])/2^(n-1)], {n, 0, 12}, {k, 0, n}]//Flatten
%o A320085 (Maxima)
%o A320085 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 A320085 tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, n))$
%o A320085 (Sage)
%o A320085 def A320085(n,k): return numerator(n*(binomial(n-1, k-1) - binomial(n-1, k))/2^(n-1))
%o A320085 flatten([[A320085(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jul 19 2021
%Y A320085 Inspired by A141692.
%Y A320085 Cf. A007318, A128433, A128434, A319861, A319862, A320086.
%K A320085 sign,easy,tabl,frac
%O A320085 0,7
%A A320085 _Franck Maminirina Ramaharo_, Oct 05 2018