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.
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, 64, 64, 64, 64, 64, 64, 64, 16, 8, 8, 8, 1, 8, 8, 8, 16, 256, 256, 64, 64, 128, 128, 64, 64, 256, 256, 256, 32, 256, 16, 128, 1, 128, 16, 256, 32, 256
Offset: 0
Examples
Triangle begins:
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, 64, 64, 64, 64, 64, 64, 64;
16, 8, 8, 8, 1, 8, 8, 8, 16;
256, 256, 64, 64, 128, 128, 64, 64, 256, 256;
256, 32, 256, 16, 128, 1, 128, 16, 256, 32, 256;
...
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
- Rita T. Farouki, The Bernstein polynomial basis: A centennial retrospective, Computer Aided Geometric Design Vol. 29 (2012), 379-419.
- Ron Goldman, Pyramid Algorithms. A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling, Morgan Kaufmann Publishers, 2002, Chap. 5.
- Eric Weisstein's World of Mathematics, Bernstein Polynomial
- Wikipedia, Bernstein polynomial
Programs
-
Maple
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
-
Mathematica
Table[Denominator[n*(Binomial[n-1, k-1] - Binomial[n-1, k])/2^(n-1)], {n, 0, 12}, {k, 0, n}]//Flatten -
Maxima
T(n, k) := 2^(n - 1)/gcd(n*(binomial(n - 1, k - 1) - binomial(n - 1, k)), 2^(n - 1))$ tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, n))$
-
Sage
def A320086(n,k): return denominator(n*(binomial(n-1, k-1) - binomial(n-1, k))/2^(n-1)) flatten([[A320086(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 19 2021