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 A127675 #11 Aug 29 2019 17:39:51
%S A127675 1,-4,3,16,-20,5,-64,112,-56,7,256,-576,432,-120,9,-1024,2816,-2816,
%T A127675 1232,-220,11,4096,-13312,16640,-9984,2912,-364,13,-16384,61440,
%U A127675 -92160,70400,-28800,6048,-560,15,65536,-278528,487424,-452608,239360,-71808,11424,-816,17,-262144,1245184
%N A127675 Coefficient table for Chebyshev's U(2*n,x) polynomials in decreasing powers of (1-x^2).
%C A127675 This table gives therefore sin((2*n+1)*phi) in terms of falling odd powers of sin(phi).
%C A127675 The unsigned triangle with reversed rows is A084930 (the signs differ).
%H A127675 W. Lang, <a href="/A127675/a127675.txt">First 15 rows and more.</a>
%F A127675 a(n,m)=0 if n < m else a(n,m) = ((-4)^(n-m))*binomial(2n-m,m)*(2*n+1)/(2*(n-m)+1), n >= m >= 0. (Proof from the differential eq. for U(2*n,x): (1-x^2)*(d^2/dx^2)U(2*n,x) - 3*x*(d/dx)U(2*n,x) + 4*n*(n+1)*U(2*n,x) = 0.)
%F A127675 a(n,m)=0 if n < m else a(n,m) = Sum_{k=0..n-m} (binomial(m+k,k)*binomial(2*n+1,2*(m+k))*(-1)^(n-m)) (from de Moivre's formula for sin((2*n+1)*phi) after replacing cos(phi)^2 with 1-sin(phi)^2).
%e A127675 [1];[ -4,3];[16,-20,5];[ -64,112,-56,7];[256,-576,432,-120,9]; ...
%e A127675 Row n=3: -64*(1-x^2)^3+ 112*(1-x^2)^2 -56*(1-x^2)^1 + 7 = 64*x^6 - 80*x^4 + 24* x^2 -1 =U(6,x).
%e A127675 Row n=3: sin(7*phi)=-64*sin(phi)^7 + 112*sin(phi)^5 - 56*sin(phi)^3 + 7*sin(phi).
%Y A127675 Row sums (signed) A033999(n)=(-1)^n. Row sums (unsigned) A002315(n).
%Y A127675 Cf. A082985 (scaled coefficient table).
%K A127675 sign,tabl,easy
%O A127675 0,2
%A A127675 _Wolfdieter Lang_, Mar 07 2007