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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.
4
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
0,2
LINKS
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.
Eric Weisstein's World of Mathematics, Bernstein Polynomial
FORMULA
T(n, k) = denominator of binomial(n,k)/2^n.
T(n, k) = 2^n/A082907(n,k).
A319862(n, k)/T(n, k) = binomial(n,k)/2^n.
T(n, n-k) = T(n, k).
T(n, 0) = 2^n.
T(n, 1) = A075101(n).
EXAMPLE
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;
...
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
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}]];
PROG
(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))$
(GAP) Flat(List([0..11], n->List([0..n], k->2^n/Gcd(Binomial(n, k), 2^n)))); # Muniru A Asiru, Sep 30 2018
(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
CROSSREFS
KEYWORD
nonn,easy,frac,tabl
AUTHOR
STATUS
approved