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

Author

Franck Maminirina Ramaharo, Sep 29 2018

Keywords

• nonn
• easy
• tabl
• frac

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.

Examples

			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;
  ...
		

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

Crossrefs

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

Programs

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

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}]];

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))$

Sage

flatten([[numerator(binomial(n,k)/2^n) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 19 2021

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).