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%I #43 Aug 17 2018 21:31:20
%S 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 32,24,-6,-1,128,-64,-192,80,80,-24,-8,1,256,-128,-448,192,240,-80,
%U -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 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 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 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 J. C. Mason and D. C. Handscomb, Chebyshev polynomials, Chapman and Hall/CRC, 2002.
%H 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 V(n+1,x) = 2xV(n,x) - V(n-1,x) with V(0,x) = 1, V(1,x) = 2x-1.
%F From _Peter Bala_, Jan 17 2014: (Start)
%F 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 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 V(n,x) = U(n,x) - U(n-1,x);
%F V(n,x) + V(n-1,x) = 2*T(n,x);
%F V(n,x) = 1/u*T(2*n+1,u) with u = sqrt((1 + x)/2).
%F Also binomial(2*n,n)*V(n,x) = 2^(2*n)*Jacobi_P(n,-1/2,1/2,x). (End)
%e 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 Triangle begins:
%e 1;
%e 2, -1;
%e 4, -2, -1;
%e 8, -4, -4, 1;
%e 16, -8, -12, 4, 1;
%e 32, -16, -32, 12, 6, -1;
%e 64, -32, -80, 32, 24, -6, -1;
%e 128, -64, -192, 80, 80, -24, -8, 1;
%e 256, -128, -448, 192, 240, -80, -40, 8, 1;
%e 512, -256, -1024, 448, 672, -240, -160, 40, 10, -1;
%e 1024, -512, -2304, 1024, 1792, -672, -560, 160, 60, -10, -1;
%e ...
%p A228565 := proc(n,k)
%p local t,Vn,x ;
%p t := arccos(x) ;
%p Vn := cos((n+1/2)*t)/cos(t/2) ;
%p coeftayl(%,x=0,n-k) ;
%p end proc:
%p for n from 0 to 10 do
%p for k from 0 to n do
%p printf("%d,",A228565(n,k)) ;
%p end do:
%p printf("\n") ;
%p end do: # _R. J. Mathar_, Mar 12 2014
%t V[n_] := Cos[(2*n + 1)*(ArcCos[x]/2)]/Cos[ArcCos[x]/2];
%t row[n_] := CoefficientList[V[n] + O[x]^(n + 1), x] // Reverse;
%t Table[row[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Nov 20 2017 *)
%Y Cf. A180870, A028297. A008312, A155751.
%K easy,tabl,sign
%O 0,2
%A _Jonny Griffiths_, Aug 25 2013
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