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A219235
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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.
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1
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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
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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.
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CROSSREFS
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Cf. A111125 (tau(n,x) coefficients if signed).
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KEYWORD
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sign,tabf
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AUTHOR
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STATUS
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approved
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