login
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.
2
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, 3, 3, -7, -35, -63, -35, 35, 63, 35, 7, -1, -3, -7, -7, 0, 7, 7, 3, 1, -9, -63, -45, -63, -63, 63, 63, 45, 63, 9, -5, -5, -135, -15, -105, 0, 105, 15, 135, 5, 5
OFFSET
0,7
COMMENTS
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).
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) = numerator of 2*A141692(n,k)/A000079(n).
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)).
T(n, n-k) = -T(n,k).
T(n, 0) = -n.
T(2*n+1, 1) = -A000466(n).
T(2*n, 1) = -A069834(n-1), n > 1.
T(n, k)/A320086(n,k) = 4*n*(k/n - 1/2)*A319861(n,k)/A319861(n,k).
Sum_{k=0..n} k*T(n,k)/A320086(n,k) = n.
Sum_{k=0..n} k^2*T(n,k)/A320086(n,k) = n^2.
Sum_{k=0..n} k*(k-1)*T(n,k)/A320086(n,k) = n*(n - 1).
EXAMPLE
Triangle begins:
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, 3, 3;
-7, -35, -63, -35, 35, 63, 35, 7;
-1, -3, -7, -7, 0, 7, 7, 3, 1;
-9, -63, -45, -63, -63, 63, 63, 45, 63, 9;
-5, -5, -135, -15, -105, 0, 105, 15, 135, 5, 5;
...
MAPLE
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
MATHEMATICA
Table[Numerator[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) := n*(binomial(n - 1, k - 1) - binomial(n - 1, k))/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 A320085(n, k): return numerator(n*(binomial(n-1, k-1) - binomial(n-1, k))/2^(n-1))
flatten([[A320085(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 19 2021
CROSSREFS
KEYWORD
sign,easy,tabl,frac
AUTHOR
STATUS
approved