A220667 - OEIS

%I #4 Dec 18 2012 17:16:20

%S 8,0,0,0,1,-8,0,12,0,-6,0,1,0,0,0,-27,0,27,0,-9,0,1,8,0,-48,0,108,0,

%T -112,0,54,0,-12,0,1,0,0,0,125,0,-375,0,450,0,-275,0,90,0,-15,0,1,-8,

%U 0,108,0,-558,0,1389,0,-1782,0,1287,0,-546,0,135,0,-18,0,1,0,0,0,-343,0,2058,0,-5145,0,7007,0,-5733,0,2940,0,-952,0,189,0,-21,0,1

%N Coefficient array for the cube of Chebyshev's C polynomials.

%C The row lengths sequence is 3*n+1 = A016777(n).

%C For the coefficient array of C(n,x) see A127672 (where C is called R).

%C The row polynomials are C(n,x)^3 = sum(a(n,m)*x^m,m=0..3*n), n >= 0, with Chebyshev's C polynomials.

%C The o.g.f. for the row polynomials is GC3(x,z) :=  sum((C(n,x)^3)*z^n,n=0..infinity) = (8*(1+z^2) + x*z*(16-7*x^2-(x*z)^2) + 4*z^2*x^2*(x^2-3))/((1+z^2-z*x)*(1+z^2-z*x*(x^2-3))).

%F a(n,m) = [x^m] (C(n,x)^3), n >= 0, 0<= m < = 3*n, with C the  monic integer version of Chebyshev's T-polynomials.

%F a(n,m) = [x^m] ([z^n] GC3(x,z)), n >= 0, 0<= m < = 3*n, with the o.g.f. GC3 given in a comment above.

%e The array a(n,m) begins:

%e n\m  0  1   2   3    4   5     6   7    8   9    10  11  12 ...

%e 0:   8

%e 1:   0  0   0   1

%e 2:  -8  0  12   0   -6   0     1

%e 3:   0  0   0 -27    0  27     0  -9    0   1

%e 4:   8  0 -48   0  108   0  -112   0   54   0   -12   0   1

%e ...

%e Row n=5: [0, 0, 0, 125, 0, -375, 0, 450, 0, -275, 0, 90, 0, -15, 0, 1],

%e Row n=6: [-8, 0, 108, 0, -558, 0, 1389, 0, -1782, 0, 1287, 0, -546, 0, 135, 0, -18, 0, 1],

%e Row n=7: [0, 0, 0, -343, 0, 2058, 0, -5145, 0, 7007, 0, -5733, 0, 2940, 0, -952, 0, 189, 0, -21, 0, 1].

%e Row n=0:  C(0,x)^3 = 2^3 = 8.

%e Row n=1:  C(1,x)^3 = x^3.

%e Row n=2:  C(2,x)^3 = (-3 + x^2)^3 =  -8 + 12*x^2 - 6*x^4 + 1*x^6.

%Y Cf. A127672.

%K sign,easy,tabf

%O 0,1

%A _Wolfdieter Lang_, Dec 18 2012