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A319862 Triangle read by rows, 0 <= k <= n: T(n,k) is the denominator of the k-th Bernstein basis polynomial of degree n evaluated at the interval midpoint t = 1/2; numerator is A319861. 4
1, 2, 2, 4, 2, 4, 8, 8, 8, 8, 16, 4, 8, 4, 16, 32, 32, 16, 16, 32, 32, 64, 32, 64, 16, 64, 32, 64, 128, 128, 128, 128, 128, 128, 128, 128, 256, 32, 64, 32, 128, 32, 64, 32, 256, 512, 512, 128, 128, 256, 256, 128, 128, 512, 512, 1024, 512, 1024, 128, 512, 256 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

T(n,k) is the denominator of binomial(n,k)/2^n.

LINKS

Table of n, a(n) for n=0..60.

American Mathematical Society, From B├ęzier to Bernstein

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/A082907(n,k).

A319862(n,k)/T(n,k) = binomial(n,k)/2^n.

T(n,n-k) = T(n,k); T(n,0) = 2^n.

T(n,1) = A075101(n).

EXAMPLE

Triangle begins:

    1;

    2,   2;

    4,   2,   4;

    8,   8,   8,   8;

   16,   4,   8,   4,  16;

   32,  32,  16,  16,  32,  32;

   64,  32,  64,  16,  64,  32,  64;

  128, 128, 128, 128, 128, 128, 128, 128;

  256,  32,  64,  32, 128,  32,  64,  32, 256;

  512, 512, 128, 128, 256, 256, 128, 128, 512, 512;

  ...

MAPLE

a:=(n, k)->2^n/gcd(binomial(n, k), 2^n): seq(seq(a(n, k), k=0..n), n=0..11); # Muniru A Asiru, Sep 30 2018

MATHEMATICA

T[n_, k_] = 2^n/GCD[Binomial[n, k], 2^n];

tabl[nn_] = TableForm[Table[T[n, k], {n, 0, nn}, {k, 0, n}]];

PROG

(Maxima)

T(n, k) := 2^n/gcd(binomial(n, k), 2^n)$

tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, n))$

(GAP) Flat(List([0..11], n->List([0..n], k->2^n/Gcd(Binomial(n, k), 2^n)))); # Muniru A Asiru, Sep 30 2018

CROSSREFS

Cf. A007318, A082907, A128433, A128434, A319861.

Sequence in context: A217839 A083779 A045865 * A220977 A054134 A319822

Adjacent sequences:  A319859 A319860 A319861 * A319863 A319864 A319865

KEYWORD

nonn,easy,frac,tabl

AUTHOR

Franck Maminirina Ramaharo, Sep 29 2018

STATUS

approved

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Last modified July 5 21:00 EDT 2020. Contains 335473 sequences. (Running on oeis4.)