OFFSET
0,2
COMMENTS
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.
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
REFERENCES
J. C. Mason and D. C. Handscomb, Chebyshev polynomials, Chapman and Hall/CRC, 2002.
LINKS
Paul Barry, On the Group of Almost-Riordan Arrays, arXiv preprint arXiv:1606.05077 [math.CO], 2016.
FORMULA
V(n+1,x) = 2xV(n,x) - V(n-1,x) with V(0,x) = 1, V(1,x) = 2x-1.
From Peter Bala, Jan 17 2014: (Start)
O.g.f. (1 - t)/(1 - 2*x*t + t^2) = 1 + (2*x - 1)*t +(4*x^2 - 2*x - 1)*t^2 + ....
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
V(n,x) = U(n,x) - U(n-1,x);
V(n,x) + V(n-1,x) = 2*T(n,x);
V(n,x) = 1/u*T(2*n+1,u) with u = sqrt((1 + x)/2).
Also binomial(2*n,n)*V(n,x) = 2^(2*n)*Jacobi_P(n,-1/2,1/2,x). (End)
EXAMPLE
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, ...
Triangle begins:
1;
2, -1;
4, -2, -1;
8, -4, -4, 1;
16, -8, -12, 4, 1;
32, -16, -32, 12, 6, -1;
64, -32, -80, 32, 24, -6, -1;
128, -64, -192, 80, 80, -24, -8, 1;
256, -128, -448, 192, 240, -80, -40, 8, 1;
512, -256, -1024, 448, 672, -240, -160, 40, 10, -1;
1024, -512, -2304, 1024, 1792, -672, -560, 160, 60, -10, -1;
...
MAPLE
A228565 := proc(n, k)
local t, Vn, x ;
t := arccos(x) ;
Vn := cos((n+1/2)*t)/cos(t/2) ;
coeftayl(%, x=0, n-k) ;
end proc:
for n from 0 to 10 do
for k from 0 to n do
printf("%d, ", A228565(n, k)) ;
end do:
printf("\n") ;
end do: # R. J. Mathar, Mar 12 2014
MATHEMATICA
V[n_] := Cos[(2*n + 1)*(ArcCos[x]/2)]/Cos[ArcCos[x]/2];
row[n_] := CoefficientList[V[n] + O[x]^(n + 1), x] // Reverse;
Table[row[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Nov 20 2017 *)
CROSSREFS
KEYWORD
AUTHOR
Jonny Griffiths, Aug 25 2013
STATUS
approved