Search: id:A125951
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%I A125951
%S A125951 1,1,2,2,4,5,10,15,26,42,74,121,212,357,620,1064,1856,3209,5618,9794,
%T A125951 17192,30153,53114,93554,165308,292250,517802,918207,1630932,2899434,
%U A125951 5161442,9196168,16402764,29281168,52319364,93555601,167427844
%N A125951 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).
%C A125951 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?
%D A125951 T. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 
               1976, Theorem 14.8.
%F A125951 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).)
%e A125951 f(1) = f(2) = 1 because 1 - z - z^2 - z^3 = (1-z)^1 *(1-z^2)^1 * ....
%Y A125951 Sequence in context: A000014 A114851 A099364 this_sequence A054538 A095020 
               A127825
%Y A125951 Adjacent sequences: A125948 A125949 A125950 this_sequence A125952 A125953 
               A125954
%K A125951 nonn
%O A125951 1,3
%A A125951 Barry Brent (barrybrent(AT)member.ams.org), Feb 04 2007

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