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A220667
Coefficient array for the cube of Chebyshev's C polynomials.
0
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))).
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
KEYWORD
sign,easy,tabf
AUTHOR
Wolfdieter Lang, Dec 18 2012
STATUS
approved