A319861 Triangle read by rows, 0 <= k <= n: T(n,k) is the numerator of the k-th Bernstein basis polynomial of degree n evaluated at the interval midpoint t = 1/2; denominator is A319862.

1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 3, 1, 1, 1, 5, 5, 5, 5, 1, 1, 3, 15, 5, 15, 3, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 1, 7, 7, 35, 7, 7, 1, 1, 1, 9, 9, 21, 63, 63, 21, 9, 9, 1, 1, 5, 45, 15, 105, 63, 105, 15, 45, 5, 1, 1, 11, 55, 165, 165, 231, 231, 165, 165, 55, 11, 1

Offset

0,8

Comments

In Computer-Aided Geometric Design, the affine combination Sum_{k=0..n} (T(n,k)/A319862(n,k))*P_k is the halfway point for the Bézier curve of degree n defined by the control points P_k, k = 0, 1, ..., n.

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

American Mathematical Society, From Bézier 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) = numerator of binomial(n,k)/2^n.

T(n, k) = binomial(n,k)/A082907(n,k).

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

T(n, n-k) = T(n,k).

T(n, 0) = 1.

T(n, 1) = A000265(n) (with offset 0, following Peter Luschny's formula).

T(n, 2) = A069834(n-1), n > 1.

Sum_{k=0..n} 2*k*T(n,k)/A319862(n,k) = n.

Sum_{k=0..n} 2*k^2*T(n,k)/A319862(n,k) = A000217(n).

Example

Triangle begins:
  1;
  1, 1;
  1, 1,  1;
  1, 3,  3,  1;
  1, 1,  3,  1,   1;
  1, 5,  5,  5,   5,  1;
  1, 3, 15,  5,  15,  3,   1;
  1, 7, 21, 35,  35, 21,   7,  1;
  1, 1,  7,  7,  35,  7,   7,  1,  1;
  1, 9,  9, 21,  63, 63,  21,  9,  9, 1;
  1, 5, 45, 15, 105, 63, 105, 15, 45, 5, 1;
  ...

Maple

a:=(n, k)->binomial(n, k)/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_] = Binomial[n, k]/GCD[Binomial[n, k], 2^n];
tabl[nn_] = TableForm[Table[T[n, k], {n, 0, nn}, {k, 0, n}]];

Prog

(Maxima)
T(n, k) := binomial(n, k)/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->Binomial(n, k)/Gcd(Binomial(n, k), 2^n)))); # Muniru A Asiru, Sep 30 2018
(Sage) flatten([[numerator(binomial(n, k)/2^n) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 19 2021

Crossrefs

Cf. A007318, A082907, A128433, A128434, A319862.

Sequence in context: A086703 A242849 A072917 * A114266 A230206 A285117

Adjacent sequences: A319858 A319859 A319860 * A319862 A319863 A319864

Keyword

nonn,easy,tabl,frac

Author

Franck Maminirina Ramaharo, Sep 29 2018

Status

approved