This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A319861 #29 Feb 16 2025 08:33:56
%S A319861 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,
%T A319861 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,
%U A319861 63,105,15,45,5,1,1,11,55,165,165,231,231,165,165,55,11,1
%N 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.
%C A319861 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.
%H A319861 G. C. Greubel, <a href="/A319861/b319861.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A319861 American Mathematical Society, <a href="http://www.ams.org/publicoutreach/feature-column/fcarc-bezier">From Bézier to Bernstein</a>
%H A319861 Rita T. Farouki, <a href="https://doi.org/10.1016/j.cagd.2012.03.001">The Bernstein polynomial basis: A centennial retrospective</a>, Computer Aided Geometric Design Vol. 29 (2012), 379-419.
%H A319861 Ron Goldman, <a href="https://doi.org/10.1016/B978-1-55860-354-7.X5000-4">Pyramid Algorithms. A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling</a>, Morgan Kaufmann Publishers, 2002, Chap. 5.
%H A319861 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BernsteinPolynomial.html">Bernstein Polynomial</a>
%H A319861 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bernstein_polynomial">Bernstein polynomial</a>
%F A319861 T(n, k) = numerator of binomial(n,k)/2^n.
%F A319861 T(n, k) = binomial(n,k)/A082907(n,k).
%F A319861 T(n, k)/A319862(n,k) = binomial(n,k)/2^n.
%F A319861 T(n, n-k) = T(n,k).
%F A319861 T(n, 0) = 1.
%F A319861 T(n, 1) = A000265(n) (with offset 0, following _Peter Luschny_'s formula).
%F A319861 T(n, 2) = A069834(n-1), n > 1.
%F A319861 Sum_{k=0..n} 2*k*T(n,k)/A319862(n,k) = n.
%F A319861 Sum_{k=0..n} 2*k^2*T(n,k)/A319862(n,k) = A000217(n).
%e A319861 Triangle begins:
%e A319861 1;
%e A319861 1, 1;
%e A319861 1, 1, 1;
%e A319861 1, 3, 3, 1;
%e A319861 1, 1, 3, 1, 1;
%e A319861 1, 5, 5, 5, 5, 1;
%e A319861 1, 3, 15, 5, 15, 3, 1;
%e A319861 1, 7, 21, 35, 35, 21, 7, 1;
%e A319861 1, 1, 7, 7, 35, 7, 7, 1, 1;
%e A319861 1, 9, 9, 21, 63, 63, 21, 9, 9, 1;
%e A319861 1, 5, 45, 15, 105, 63, 105, 15, 45, 5, 1;
%e A319861 ...
%p A319861 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 A319861 T[n_, k_] = Binomial[n, k]/GCD[Binomial[n, k], 2^n];
%t A319861 tabl[nn_] = TableForm[Table[T[n, k], {n, 0, nn}, {k, 0, n}]];
%o A319861 (Maxima)
%o A319861 T(n,k) := binomial(n, k)/gcd(binomial(n, k), 2^n)$
%o A319861 tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, n))$
%o A319861 (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
%o A319861 (Sage) flatten([[numerator(binomial(n,k)/2^n) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jul 19 2021
%Y A319861 Cf. A007318, A082907, A128433, A128434, A319862.
%K A319861 nonn,easy,tabl,frac
%O A319861 0,8
%A A319861 _Franck Maminirina Ramaharo_, Sep 29 2018