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A125951
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Exponents f(n), n = 1, 2, ..., for the infinite product 1 -z - z^2 - z^3 =Product_{n=1}^{\infty} (1-z^n)^f(n).
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0
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1, 1, 2, 2, 4, 5, 10, 15, 26, 42, 74, 121, 212, 357, 620, 1064, 1856, 3209, 5618, 9794, 17192, 30153, 53114, 93554, 165308, 292250, 517802, 918207, 1630932, 2899434, 5161442, 9196168, 16402764, 29281168, 52319364, 93555601, 167427844
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Let w = z + z^2 + z^3. Then 1 - z - z^2 - z^3 = 1 - 1w = (by the cyclotomic identity) Product_{n=1}^{\infty} (1-w^n)^P(1,n), where P is the necklace polynomial. P is a counting function. Is f also a counting function?
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REFERENCES
| T. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, Theorem 14.8.
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FORMULA
| Let r(n) be the coefficient of z^n in 1 - z - z^2 - z^3, so that r(0) = 1 and r(n) = 0 for n>3. Let F(k) satisfy the recurrence n r(n) + sum_{k=1}^n r(n-k)F(k) = 0. Let \mu be the usual M\"obius function. Then f(n) = (1/n) sum_{d|n} \mu(n/d) F(d) (so that n*f(n) is the M\"obius inverse of F(n).)
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EXAMPLE
| f(1) = f(2) = 1 because 1 - z - z^2 - z^3 = (1-z)^1 *(1-z^2)^1 * ....
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CROSSREFS
| Sequence in context: A000014 A114851 A099364 * A054538 A095020 A127825
Adjacent sequences: A125948 A125949 A125950 * A125952 A125953 A125954
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KEYWORD
| nonn
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AUTHOR
| Barry Brent (barrybrent(AT)member.ams.org), Feb 04 2007
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