A219235 Coefficient array for the third power of the monic integer Chebyshev polynomials 2*T(2*n+1,x/2)/x as a function of x^2.

1, -27, 27, -9, 1, 125, -375, 450, -275, 90, -15, 1, -343, 2058, -5145, 7007, -5733, 2940, -952, 189, -21, 1, 729, -7290, 30861, -72927, 107406, -104652, 69768, -32319, 10395, -2277, 324, -27, 1, -1331, 19965, -127776, 461857, -1058145, 1641486, -1797818, 1427679, -834900, 361790, -115830, 27027, -4466, 495, -33, 1

Offset: 0

Author

Wolfdieter Lang, Nov 27 2012

Keywords

• sign
• tabf

Comments

The length of row n of this array is 3*n+1; see A016777.

Define tau(n,x):= C(2*n+1,x)/x, with C the monic integer Chebyshev T-polynomials with their coefficients given in A127672 (C(n,x) = 2*T(n,x/2) is there called R(n,x)). The coefficients of the x^2-powers of tau(n) are found as signed A111125 (see the last of the comments from Oct 18 2012 there). The irregular triangle a(n,m) appears in tau(n,x)^3 = sum(a(n,m)*x(2*m),m=0..3*n), n>=0.

The o.g.f. of the row polynomials as function of x^2 is G(3;x,z) := sum(tau(n,x)^3*z^n, n=0..infinity) =

(1 - (23-17*x^2+3*x^4)*z*(1-z) - z^3)/(((z+1)^2-x^2*z)*((z+1)^2-z*x^2*(x^2-3)^2)). From the odd part of the bisection of the o.g.f. for C(n,x)^3 divided by x^3.

Examples

			The array a(n,m) begins:
n\m   0    1     2    3     4     5     6   7   8  9 ...
0:    1
1:  -27   27    -9    1
2:  125 -375   450 -275    90   -15     1
3: -343 2058 -5145 7007 -5733  2940  -952 189 -21  1
...
Row n=4: [729, -7290, 30861, -72927, 107406, -104652, 69768, -32319, 10395, -2277, 324, -27, 1].
Row n=5: [-1331, 19965, -127776, 461857, -1058145, 1641486, -1797818, 1427679, -834900, 361790, -115830, 27027, -4466, 495, -33, 1].
Row n=1 polynomial p(1,x) := -27 + 27*x - 9*x^2 + 1*x^3 with p(1,x^2) = tau(1,x)^3 = (-3 + x^2)^3 = -27+27*x^2-9*x^4+x^6.
		

Crossrefs

Cf. A111125 (tau(n,x) coefficients if signed).

Formula

a(n,m) = [x^(2*m)] tau(n,x)^3, n>=0, m=0,1,...,3*n, with the monic integer polynomials tau(n,x) defined above in a comment.