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.
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
KEYWORD
AUTHOR
Franck Maminirina Ramaharo, Sep 29 2018
STATUS
approved