A319862 Triangle read by rows, 0 <= k <= n: T(n,k) is the denominator of the k-th Bernstein basis polynomial of degree n evaluated at the interval midpoint t = 1/2; numerator is A319861.
1, 2, 2, 4, 2, 4, 8, 8, 8, 8, 16, 4, 8, 4, 16, 32, 32, 16, 16, 32, 32, 64, 32, 64, 16, 64, 32, 64, 128, 128, 128, 128, 128, 128, 128, 128, 256, 32, 64, 32, 128, 32, 64, 32, 256, 512, 512, 128, 128, 256, 256, 128, 128, 512, 512, 1024, 512, 1024, 128, 512, 256, 512, 128, 1024, 512, 1024
Offset: 0
Examples
Triangle begins:
1;
2, 2;
4, 2, 4;
8, 8, 8, 8;
16, 4, 8, 4, 16;
32, 32, 16, 16, 32, 32;
64, 32, 64, 16, 64, 32, 64;
128, 128, 128, 128, 128, 128, 128, 128;
256, 32, 64, 32, 128, 32, 64, 32, 256;
512, 512, 128, 128, 256, 256, 128, 128, 512, 512;
...
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
- American Mathematical Society, From Bézier to Bernstein
- 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
-
GAP
Flat(List([0..11],n->List([0..n],k->2^n/Gcd(Binomial(n,k),2^n)))); # Muniru A Asiru, Sep 30 2018
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Maple
a:=(n,k)->2^n/gcd(binomial(n,k),2^n): seq(seq(a(n,k),k=0..n),n=0..11); # Muniru A Asiru, Sep 30 2018
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Mathematica
T[n_, k_] = 2^n/GCD[Binomial[n, k], 2^n]; tabl[nn_] = TableForm[Table[T[n, k], {n, 0, nn}, {k, 0, n}]]; -
Maxima
T(n, k) := 2^n/gcd(binomial(n, k), 2^n)$ tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, n))$
-
Sage
def A319862(n,k): return denominator(binomial(n,k)/2^n) flatten([[A319862(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 20 2021