A219240 Coefficient array for the cube of Chebyshev's S polynomials.

1, 0, 0, 0, 1, -1, 0, 3, 0, -3, 0, 1, 0, 0, 0, -8, 0, 12, 0, -6, 0, 1, 1, 0, -9, 0, 30, 0, -45, 0, 30, 0, -9, 0, 1, 0, 0, 0, 27, 0, -108, 0, 171, 0, -136, 0, 57, 0, -12, 0, 1, -1, 0, 18, 0, -123, 0, 399, 0, -651, 0, 588, 0, -308, 0, 93, 0, -15, 0, 1, 0, 0, 0, -64, 0, 480, 0, -1488, 0, 2488, 0, -2472, 0, 1524, 0, -588, 0, 138, 0, -18, 0, 1

Offset: 0

Author

Wolfdieter Lang, Dec 12 2012

Keywords

• sign
• tabf
• easy

Comments

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

For the coefficient triangle for Chebyshev's S polynomials see A049310.

The o.g.f. for S(n,x)^3, n >= 0, is GS(3;x,z) = (1+z^2+2*z*x)/ ((1+z^2-z*x)*(1+z^2-z*x*(x^2-3))). This is obtained from the de Moivre-Binet formula for S(n,x) and the binomial theorem.

In general the monic integer Chebyshev polynomial tau(n,x):= R(2*n+1,x)/x enters, where R(n,x) = 2*T(n,x/2) with Chebyshev's T polynomial (for R see A127672), and the coefficient triangle for tau is given in A111125 (here for the third power of S only tau(0,x) = 1 and tau(1,x) = x^2 - 3 enter).

Examples

			The array a(n,m) begins:
n\m   0  1  2  3  4    5   6    7  8    9 10  11 12  13 14 15
n=0:  1
n=1:  0  0  0  1
n=2: -1  0  3  0 -3    0   1
n=3:  0  0  0 -8  0   12   0  -6   0    1
n=4:  1  0 -9  0 30    0 -45   0  30    0 -9   0  1
n=5:  0  0  0 27  0 -108   0  171  0 -136  0  57  0 -12  0  1
...
Row n=6: [-1, 0, 18, 0, -123, 0, 399, 0, -651, 0, 588, 0, -308, 0, 93, 0, -15, 0, 1],
Row n=7: [0, 0, 0, -64, 0, 480, 0, -1488, 0, 2488, 0, -2472, 0, 1524, 0, -588, 0, 138, 0, -18, 0, 1],
Row n=8: [1, 0, -30, 0, 345, 0, -1921, 0, 5598, 0, -9540, 0, 10212, 0, -7137, 0, 3303, 0, -1003, 0, 192, 0, -21, 0, 1].
n=2: S(2,x)^3 = (x^2 - 1)^3 = -1 + 3*x^2 - 3*x^4 + x^6.
n=3: S(3,x)^3 = (x^3 - 2*x)^3 = -8*x^3 + 12*x^5 - 6*x^7 + x^9.
		

Crossrefs

Cf. A049310, A127672, A158454 (square of S polynomials), A219234 (fourth power of S polynomials).

Formula

a(n,m) = [x^m] S(n, x)^3, n >= 0, 0 <= m <= 3*n, with Chebyshev's S polynomials (see A049310).

a(n,m) = [x^m]([z^n] GS(3;x,z)), with the o.g.f. GS(3;x,z) given above in a comment.

The row polynomials p(n, x) := Sum_{m=0..3*n} a(n,m)*x^m = S(n, x)^3 are (S(3*n+2, x) - 3*S(n, x))/(x^2 - 4). For the factorization of S polynomials see comments on A049310. - Wolfdieter Lang, Apr 09 2018