A220667 Coefficient array for the cube of Chebyshev's C polynomials.

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

Offset

0,1

Comments

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

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

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.

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))).

Links

Table of n, a(n) for n=0..91.

Formula

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.

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.

Example

The array a(n,m) begins:
n\m  0  1   2   3    4   5     6   7    8   9    10  11  12 ...
0:   8
1:   0  0   0   1
2:  -8  0  12   0   -6   0     1
3:   0  0   0 -27    0  27     0  -9    0   1
4:   8  0 -48   0  108   0  -112   0   54   0   -12   0   1
...
Row n=5: [0, 0, 0, 125, 0, -375, 0, 450, 0, -275, 0, 90, 0, -15, 0, 1],
Row n=6: [-8, 0, 108, 0, -558, 0, 1389, 0, -1782, 0, 1287, 0, -546, 0, 135, 0, -18, 0, 1],
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].
Row n=0:  C(0,x)^3 = 2^3 = 8.
Row n=1:  C(1,x)^3 = x^3.
Row n=2:  C(2,x)^3 = (-3 + x^2)^3 =  -8 + 12*x^2 - 6*x^4 + 1*x^6.

Crossrefs

Cf. A127672.

Sequence in context: A325737 A272625 A334709 * A317445 A199619 A036482

Adjacent sequences: A220664 A220665 A220666 * A220668 A220669 A220670

Keyword

sign,easy,tabf

Author

Wolfdieter Lang, Dec 18 2012

Status

approved