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A220668 Coefficient array for the powers of x^2 of the square of the even-indexed Chebyshev C polynomials.

%I A220668 #10 Dec 10 2016 21:44:55
%S A220668 4,4,-4,1,4,-16,20,-8,1,4,-36,105,-112,54,-12,1,4,-64,336,-672,660,
%T A220668 -352,104,-16,1,4,-100,825,-2640,4290,-4004,2275,-800,170,-20,1,4,
%U A220668 -144,1716,-8008,19305,-27456,24752,-14688,5814,-1520,252,-24,1,4,-196,3185,-20384,68068,-136136,176358,-155040,94962,-40964,12397,-2576,350,-28,1
%N A220668 Coefficient array for the powers of x^2 of the square of the even-indexed Chebyshev C polynomials.
%C A220668 The row lengths sequence of this irregular triangle is 2*n + 1 = A005408(n), n>=0.
%C A220668 For the coefficient triangle for Chebyshev's C polynomials see A127672 (where they are called R polynomials).
%C A220668 a(n,m) is the coefficient of (x^2)^m of C(2*n,x)^2. The o.g.f. for the row polynomials P(n,x) = sum(a(n,m)*x^m,m=0..2*n) is GC2even(x,z) := sum( P(n,x)*z^n,n=0..infinity) =
%C A220668 (4 - (8 - 12*x + 3*x^2)*z + (x - 2)^2*z^2)/((1 - z)*(1 - ((x-2)^2 - 2)*z + z^2)). From the even part of the bisection of the o.g.f. for the square of the C polynomials.
%F A220668 a(n,m) = [x^m] C(n,x)^2, n >= 0, 0 <= m <= 2*n, with Chebyshev's C polynomials (see A127672).
%F A220668 a(n,m) =[x^m]([z]^n GC2even(x,z)), with the o.g.f. GC2even(x,z) given in a comment above.
%e A220668 The array begins:
%e A220668 n\m 0     1    2      3     4     5     6     7    8    9  10
%e A220668 0:  4
%e A220668 1:  4    -4    1
%e A220668 2:  4   -16   20     -8     1
%e A220668 3:  4   -36  105   -112    54   -12     1
%e A220668 4:  4   -64  336   -672   660  -352   104   -16    1
%e A220668 5:  4  -100  825  -2640  4290 -4004  2275  -800  170  -20   1
%e A220668 ...
%e A220668 Row 6: [4, -144, 1716, -8008, 19305, -27456, 24752, -14688, 5814, -1520, 252, -24, 1],
%e A220668 Row 7: [4, -196, 3185, -20384, 68068, -136136, 176358, -155040, 94962, -40964, 12397, -2576, 350, -28, 1].
%e A220668 Row n=2: C(2,x)^2 = (-2 + x^2)^2 = 4 - 4*x^2 + 1*x^4, with
%e A220668 the row polynomial P(2,x) = C(2,sqrt(x))^2 = 4 - 4*x + 1*x^2.
%Y A220668 Cf. A127672.
%K A220668 sign,easy,tabf
%O A220668 0,1
%A A220668 _Wolfdieter Lang_, Dec 26 2012