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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, 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, 1024, 1024 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

T(n,k) is the denominator of 2*A141692(n,k)/A000079(n).

LINKS

Table of n, a(n) for n=0..67.

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) = 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[Table[Denominator[n*(Binomial[n - 1, k - 1] - Binomial[n - 1, k])/2^(n - 1)], {k, 0, n}], {n, 0, 12}]//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))$

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

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Last modified September 19 12:57 EDT 2019. Contains 327198 sequences. (Running on oeis4.)