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.

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

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) = 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

Inspired by A141692.

Cf. A007318, A128433, A128434, A319861, A319862, A320085.

Sequence in context: A172369 A190338 A199177 * A074803 A177229 A046595

Adjacent sequences: A320083 A320084 A320085 * A320087 A320088 A320089

Keyword

nonn,tabl,easy,frac

Author

Franck Maminirina Ramaharo, Oct 05 2018

Status

approved