login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 4
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 (list; table; graph; refs; listen; history; text; internal format)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 12 23:52 EDT 2022. Contains 356077 sequences. (Running on oeis4.)