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A302707
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Number of factors of Chebyshev polynomial S(2*n+1, x) (A049310) over the integers. Factorization is into the minimal integer polynomials C (A187360).
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2
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1, 2, 4, 3, 4, 6, 4, 4, 7, 6, 4, 8, 4, 6, 10, 5, 4, 10, 4, 8, 10, 6, 4, 10, 7, 6, 10, 8, 4, 14, 4, 6, 10, 6, 10, 13, 4, 6, 10, 10, 4, 14, 4, 8, 16, 6, 4, 12, 7, 10, 10, 8, 4, 14, 10, 10, 10, 6, 4, 18, 4, 6, 16, 7, 10, 14, 4, 8, 10, 14, 4, 16, 4, 6, 16, 8, 10, 14, 4, 12, 13
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OFFSET
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0,2
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COMMENTS
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For the factorization of the Chebyshev S polynomials (coefficients in A049310) with odd index into the minimal polynomials of {2*cos(Pi/k)}_{k>=1} (coefficients in A187360) see an Apr 12 2018 comment in A049310.
Note that factors -C(k, -x) may appear also and they come always together with C(k, x) (the minus signs are not counted as factors here). C(2, x) = x is always a factor.
For the number of factors of S(2*n, x) see 2*(tau(2*n+1) - 1) = 2*A095374(n).
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LINKS
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FORMULA
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a(n) = tau_{odd}(n+1) + tau(2*(n+1)) - 2, n >= 0, with tau_{odd} = A001227 and tau = A000005.
G.f.: Sum_{k>=1} (x^(k-1)/(1-x^(2*k)) + x^(k-1)*(2+x^k)/(1-x^(2*k))) - 2/(1-x).
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EXAMPLE
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a(2) = 4 because S(5, x) = 3*x-4*x^3+x^5 = x*(-1 + x)*(1 + x)*(-3 + x^2) = C(2, x)*C(3, x)*(-C(3, -x))*C(6, x).
a(5) = 6 because S(11, x) = -6*x + 35*x^3 - 56*x^5 + 36*x^7 - 10*x^9 + x^11 = x*(-1 + x)*(1 + x)*(-2 + x^2)*(-3 + x^2)*(1 - 4*x^2 + x^4) = C(2, x)*C(3, x)*(-C(3, -x))*C(4, x)*C(6, x)*C(12, x).
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PROG
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(PARI)
A001227(n) = numdiv(n>>valuation(n, 2));
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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