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A220668
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Coefficient array for the powers of x^2 of the square of the even-indexed Chebyshev C polynomials.
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1
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4, 4, -4, 1, 4, -16, 20, -8, 1, 4, -36, 105, -112, 54, -12, 1, 4, -64, 336, -672, 660, -352, 104, -16, 1, 4, -100, 825, -2640, 4290, -4004, 2275, -800, 170, -20, 1, 4, -144, 1716, -8008, 19305, -27456, 24752, -14688, 5814, -1520, 252, -24, 1, 4, -196, 3185, -20384, 68068, -136136, 176358, -155040, 94962, -40964, 12397, -2576, 350, -28, 1
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OFFSET
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0,1
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COMMENTS
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The row lengths sequence of this irregular triangle is 2*n + 1 = A005408(n), n>=0.
For the coefficient triangle for Chebyshev's C polynomials see A127672 (where they are called R polynomials).
a(n,m) is the coefficient of (x^2)^m of C(2*n,x)^2. The o.g.f. for the row polynomials P(n,x) = sum(a(n,m)*x^m,m=0..2*n) is GC2even(x,z) := sum( P(n,x)*z^n,n=0..infinity) =
(4 - (8 - 12*x + 3*x^2)*z + (x - 2)^2*z^2)/((1 - z)*(1 - ((x-2)^2 - 2)*z + z^2)). From the even part of the bisection of the o.g.f. for the square of the C polynomials.
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LINKS
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FORMULA
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a(n,m) = [x^m] C(n,x)^2, n >= 0, 0 <= m <= 2*n, with Chebyshev's C polynomials (see A127672).
a(n,m) =[x^m]([z]^n GC2even(x,z)), with the o.g.f. GC2even(x,z) given in a comment above.
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EXAMPLE
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The array begins:
n\m 0 1 2 3 4 5 6 7 8 9 10
0: 4
1: 4 -4 1
2: 4 -16 20 -8 1
3: 4 -36 105 -112 54 -12 1
4: 4 -64 336 -672 660 -352 104 -16 1
5: 4 -100 825 -2640 4290 -4004 2275 -800 170 -20 1
...
Row 6: [4, -144, 1716, -8008, 19305, -27456, 24752, -14688, 5814, -1520, 252, -24, 1],
Row 7: [4, -196, 3185, -20384, 68068, -136136, 176358, -155040, 94962, -40964, 12397, -2576, 350, -28, 1].
Row n=2: C(2,x)^2 = (-2 + x^2)^2 = 4 - 4*x^2 + 1*x^4, with
the row polynomial P(2,x) = C(2,sqrt(x))^2 = 4 - 4*x + 1*x^2.
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CROSSREFS
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KEYWORD
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sign,easy,tabf
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AUTHOR
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STATUS
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approved
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