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

8, -8, 12, -6, 1, 8, -48, 108, -112, 54, -12, 1, -8, 108, -558, 1389, -1782, 1287, -546, 135, -18, 1, 8, -192, 1776, -8032, 19308, -27456, 24752, -14688, 5814, -1520, 252, -24, 1, -8, 300, -4350, 31045, -119370, 277137, -419900, 436050, -319770, 168245, -63756, 17250, -3250, 405, -30, 1

0,1

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

The monic integer Chebyshev T-polynomials are C(n,x) := 2*T(n,x/2) (see A127672, were C is called R). The irregular triangle a(n,m) appears in C(2*n,x)^3 = sum(a(n,m)*x^(2*m), m=0..3*n), n >= 0.

The o.g.f. Ge(3;x,z) := sum(C(2*n,x)^3*z^n, n=0..infinity) =

(8 + (24-68*x^2+42*x^4-7*x^6)*z + (24-80*x^2+84*x^4-32*x^6+4*x^8)*z^2 + (8-12*x^2+6*x^4-x^6)*z^3)/(((z+1)^2-z*x^2)*((z+1)^2-z*x^2*(x^2-3)^2))).

a(n,m) = [x^(2*m)] C(2*n,x)^3, with the C-polynomials defined from Chebyshev's T-polynomials in a comment above.

The irregular triangle a(n,m) begins:

n\m 0 1 2 3 4 5 6 7 8 9 ...

O: 8

1: -8 12 -6 1

2: 8 -48 108 -112 54 -12 1

3: -8 108 -558 1389 -1782 1287 -546 135 -18 1

...

Row n=4: [8, -192, 1776, -8032, 19308, -27456, 24752, -14688, 5814, -1520, 252, -24, 1].

Row n=5: [-8, 300, -4350, 31045, -119370, 277137, -419900, 436050, -319770, 168245, -63756, 17250, -3250, 405, -30, 1].

Row n=1 polynomial p(n,1) = -8 + 12*x - 6*x^2 + 1*x^3 = C(2,sqrt(x))^3 = (-2+x)^3.

Cf. A219235 (C(2*n+1,sqrt(x))/sqrt(x))^3.

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Wolfdieter Lang, Nov 28 2012

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