login
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.
2
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
0,7
LINKS
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 2*A141692(n,k)/A000079(n).
T(n, k) = 2^(n-1)/gcd(n*(binomial(n-1, k-1) - binomial(n-1, k)), 2^(n-1)).
T(n, n-k) = T(n,k).
T(n, 0) = A084623(n), n > 0.
T(2*n+1, 1) = A000302(n).
EXAMPLE
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;
...
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
PROG
(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
CROSSREFS
KEYWORD
nonn,tabl,easy,frac
AUTHOR
STATUS
approved