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 A228565 #43 Aug 17 2018 21:31:20
%S A228565 1,2,-1,4,-2,-1,8,-4,-4,1,16,-8,-12,4,1,32,-16,-32,12,6,-1,64,-32,-80,
%T A228565 32,24,-6,-1,128,-64,-192,80,80,-24,-8,1,256,-128,-448,192,240,-80,
%U A228565 -40,8,1,512,-256,-1024,448,672,-240,-160,40,10,-1,1024,-512,-2304,1024,1792,-672,-560,160,60,-10,-1,2048,-1024,-5120,2304,4608,-1792,-1792,560,280,-60,-12,1,4096,-2048,-11264,5120,11520,-4608,-5376,1792,1120,-280,-84,12,1
%N A228565 Triangle read by rows: coefficients of descending powers of the polynomial V(n,x) = cos((2n+1)(arccos(x)/2))/cos(arccos(x)/2), n >= 0.
%C A228565 V(n,x) is related to the Dirichlet kernel and its associated polynomials. V(n,x) arises in studying recurrences connecting the Chebyshev polynomials of the first and second kinds. It differs from A180870 above only in the signs of terms.
%C A228565 Chebyshev polynomials V(n,x) of the third kind (see, for example, Mason and Handscomb, Chapter 1, Definition 1.3). See A180870 for Chebyshev polynomials of the fourth kind. Cf. A155751. - _Peter Bala_, Jan 17 2014
%D A228565 J. C. Mason and D. C. Handscomb, Chebyshev polynomials, Chapman and Hall/CRC, 2002.
%H A228565 Paul Barry, <a href="https://arxiv.org/abs/1606.05077">On the Group of Almost-Riordan Arrays</a>, arXiv preprint arXiv:1606.05077 [math.CO], 2016.
%F A228565 V(n+1,x) = 2xV(n,x) - V(n-1,x) with V(0,x) = 1, V(1,x) = 2x-1.
%F A228565 From _Peter Bala_, Jan 17 2014: (Start)
%F A228565 O.g.f. (1 - t)/(1 - 2*x*t + t^2) = 1 + (2*x - 1)*t +(4*x^2 - 2*x - 1)*t^2 + ....
%F A228565 In terms of the Chebyshev polynomials T(n,x) of the first kind and Chebyshev polynomials U(n,x) of the second kind we have
%F A228565 V(n,x) = U(n,x) - U(n-1,x);
%F A228565 V(n,x) + V(n-1,x) = 2*T(n,x);
%F A228565 V(n,x) = 1/u*T(2*n+1,u) with u = sqrt((1 + x)/2).
%F A228565 Also binomial(2*n,n)*V(n,x) = 2^(2*n)*Jacobi_P(n,-1/2,1/2,x). (End)
%e A228565 V(0,x) = 1, V(1,x) = 2x-1, V(2,x) = 4x^2-2x-1, V(3,x) = 8x^3 -4x^2 - 4x + 1, V(4,x) = 16x^4 - 8x^3 - 12x^2 + 4x + 1, V(5,x) = 32x^5 - 16x^4 - 32x^3 + 12x^2 + 6x - 1, V(6,x) =64x^6 - 32x^5 - 80x^4 + 32x^3 + 24x^2 - 6x - 1, ...
%e A228565 Triangle begins:
%e A228565 1;
%e A228565 2, -1;
%e A228565 4, -2, -1;
%e A228565 8, -4, -4, 1;
%e A228565 16, -8, -12, 4, 1;
%e A228565 32, -16, -32, 12, 6, -1;
%e A228565 64, -32, -80, 32, 24, -6, -1;
%e A228565 128, -64, -192, 80, 80, -24, -8, 1;
%e A228565 256, -128, -448, 192, 240, -80, -40, 8, 1;
%e A228565 512, -256, -1024, 448, 672, -240, -160, 40, 10, -1;
%e A228565 1024, -512, -2304, 1024, 1792, -672, -560, 160, 60, -10, -1;
%e A228565 ...
%p A228565 A228565 := proc(n,k)
%p A228565 local t,Vn,x ;
%p A228565 t := arccos(x) ;
%p A228565 Vn := cos((n+1/2)*t)/cos(t/2) ;
%p A228565 coeftayl(%,x=0,n-k) ;
%p A228565 end proc:
%p A228565 for n from 0 to 10 do
%p A228565 for k from 0 to n do
%p A228565 printf("%d,",A228565(n,k)) ;
%p A228565 end do:
%p A228565 printf("\n") ;
%p A228565 end do: # _R. J. Mathar_, Mar 12 2014
%t A228565 V[n_] := Cos[(2*n + 1)*(ArcCos[x]/2)]/Cos[ArcCos[x]/2];
%t A228565 row[n_] := CoefficientList[V[n] + O[x]^(n + 1), x] // Reverse;
%t A228565 Table[row[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Nov 20 2017 *)
%Y A228565 Cf. A180870, A028297. A008312, A155751.
%K A228565 easy,tabl,sign
%O A228565 0,2
%A A228565 _Jonny Griffiths_, Aug 25 2013