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.

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

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

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

Inspired by A141692.

Cf. A007318, A128433, A128434, A319861, A319862, A320086.

Sequence in context: A132973 A107760 A180560 * A358691 A172368 A138070

Adjacent sequences: A320082 A320083 A320084 * A320086 A320087 A320088

Keyword

sign,easy,tabl,frac

Author

Franck Maminirina Ramaharo, Oct 05 2018

Status

approved