A319862 -id:A319862 - cp's OEIS frontend

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

Franck Maminirina Ramaharo, Sep 29 2018

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.

			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;
  ...

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

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

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

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

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}]];

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))$

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

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).

A320085 Triangle read by rows, 0 <= k <= n: T(n,k) is the numerator of the derivative of the k-th Bernstein basis polynomial of degree n evaluated at the interval midpoint t = 1/2; denominator is A320086.

Original entry on oeis.org

0, -1, 1, -1, 0, 1, -3, -3, 3, 3, -1, -1, 0, 1, 1, -5, -15, -5, 5, 15, 5, -3, -3, -15, 0, 15, 3, 3, -7, -35, -63, -35, 35, 63, 35, 7, -1, -3, -7, -7, 0, 7, 7, 3, 1, -9, -63, -45, -63, -63, 63, 63, 45, 63, 9, -5, -5, -135, -15, -105, 0, 105, 15, 135, 5, 5

Offset: 0

Franck Maminirina Ramaharo, Oct 05 2018

If n = 2*k, then T(n,k) = 0 since the k-th Bernstein basis polynomial of degree n has a single unique local maximum occurring at t = k/n, which coincides with the interval midpoint t = 1/2 (T(0,0) = 0 because the only 0 degree Bernstein basis polynomial is the constant 1).

			Triangle begins:
   0;
  -1,   1;
  -1,   0,    1;
  -3,  -3,    3,   3;
  -1,  -1,    0,   1,    1;
  -5, -15,   -5,   5,   15,  5;
  -3,  -3,  -15,   0,   15,  3,   3;
  -7, -35,  -63, -35,   35, 63,  35,  7;
  -1,  -3,   -7,  -7,    0,  7,   7,  3,   1;
  -9, -63,  -45, -63,  -63, 63,  63, 45,  63, 9;
  -5,  -5, -135, -15, -105,  0, 105, 15, 135, 5, 5;
  ...

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

Inspired by A141692.

Cf. A007318, A128433, A128434, A319861, A319862, A320086.

T:=proc(n,k) n*(binomial(n-1,k-1)-binomial(n-1,k))/gcd(n*(binomial(n-1,k-1)-binomial(n-1,k)),2^(n-1)); end proc: seq(seq(T(n,k),k=0..n),n=0..11); # Muniru A Asiru, Oct 06 2018

Table[Numerator[n*(Binomial[n-1, k-1] - Binomial[n-1, k])/2^(n-1)], {n, 0, 12}, {k, 0, n}]//Flatten

T(n, k) := n*(binomial(n - 1, k - 1) - binomial(n - 1, k))/gcd(n*(binomial(n - 1, k - 1) - binomial(n - 1, k)), 2^(n - 1))$
tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, n))$

def A320085(n,k): return numerator(n*(binomial(n-1, k-1) - binomial(n-1, k))/2^(n-1))
flatten([[A320085(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 19 2021

T(n, k) = numerator of 2*A141692(n,k)/A000079(n).
T(n, k) = n*(binomial(n-1, k-1) - binomial(n-1, k))/gcd(n*(binomial(n-1, k-1) - binomial(n-1, k)), 2^(n-1)).
T(n, n-k) = -T(n,k).
T(n, 0) = -n.
T(2*n+1, 1) = -A000466(n).
T(2*n, 1) = -A069834(n-1), n > 1.
T(n, k)/A320086(n,k) = 4*n*(k/n - 1/2)*A319861(n,k)/A319861(n,k).
Sum_{k=0..n} k*T(n,k)/A320086(n,k) = n.
Sum_{k=0..n} k^2*T(n,k)/A320086(n,k) = n^2.
Sum_{k=0..n} k*(k-1)*T(n,k)/A320086(n,k) = n*(n - 1).

A320086 Triangle read by rows, 0 <= k <= n: T(n,k) is the denominator of the derivative of the k-th Bernstein basis polynomial of degree n evaluated at the interval midpoint t = 1/2; numerator is A320085.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 4, 4, 4, 4, 2, 1, 1, 1, 2, 16, 16, 8, 8, 16, 16, 16, 4, 16, 1, 16, 4, 16, 64, 64, 64, 64, 64, 64, 64, 64, 16, 8, 8, 8, 1, 8, 8, 8, 16, 256, 256, 64, 64, 128, 128, 64, 64, 256, 256, 256, 32, 256, 16, 128, 1, 128, 16, 256, 32, 256

Offset: 0

Franck Maminirina Ramaharo, Oct 05 2018

			Triangle begins:
    1;
    1,   1;
    1,   1,   1;
    4,   4,   4,  4;
    2,   1,   1,  1,   2;
   16,  16,   8,  8,  16,  16;
   16,   4,  16,  1,  16,   4,  16;
   64,  64,  64, 64,  64,  64,  64, 64;
   16,   8,   8,  8,   1,   8,   8,  8,  16;
  256, 256,  64, 64, 128, 128,  64, 64, 256, 256;
  256,  32, 256, 16, 128,   1, 128, 16, 256,  32, 256;
  ...

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

Inspired by A141692.

Cf. A007318, A128433, A128434, A319861, A319862, A320085.

T:=proc(n,k) 2^(n-1)/gcd(n*(binomial(n-1,k-1)-binomial(n-1,k)),2^(n-1)); end proc: seq(seq(T(n,k),k=0..n),n=1..11); # Muniru A Asiru, Oct 06 2018

Table[Denominator[n*(Binomial[n-1, k-1] - Binomial[n-1, k])/2^(n-1)], {n, 0, 12}, {k, 0, n}]//Flatten

T(n, k) := 2^(n - 1)/gcd(n*(binomial(n - 1, k - 1) - binomial(n - 1, k)), 2^(n - 1))$
tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, n))$

def A320086(n,k): return denominator(n*(binomial(n-1, k-1) - binomial(n-1, k))/2^(n-1))
flatten([[A320086(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 19 2021

T(n, k) = denominator of 2*A141692(n,k)/A000079(n).
T(n, k) = 2^(n-1)/gcd(n*(binomial(n-1, k-1) - binomial(n-1, k)), 2^(n-1)).
T(n, n-k) = T(n,k).
T(n, 0) = A084623(n), n > 0.
T(2*n+1, 1) = A000302(n).

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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.

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A320085 Triangle read by rows, 0 <= k <= n: T(n,k) is the numerator of the derivative of the k-th Bernstein basis polynomial of degree n evaluated at the interval midpoint t = 1/2; denominator is A320086.

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A320086 Triangle read by rows, 0 <= k <= n: T(n,k) is the denominator of the derivative of the k-th Bernstein basis polynomial of degree n evaluated at the interval midpoint t = 1/2; numerator is A320085.

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Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

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