%I #23 Nov 22 2022 20:15:36
%S 1,1,2,2,4,5,10,15,26,42,74,121,212,357,620,1064,1856,3209,5618,9794,
%T 17192,30153,53114,93554,165308,292250,517802,918207,1630932,2899434,
%U 5161442,9196168,16402764,29281168,52319364,93555601,167427844
%N Exponents f(n), n = 1, 2, ..., in the infinite product 1 - z - z^2 - z^3 = Product_{n>=1} (1-z^n)^f(n).
%C Let w = z + z^2 + z^3. Then 1 - z - z^2 - z^3 = 1 - 1w = (by the cyclotomic identity) Product_{n>=1} (1-w^n)^P(1,n), where P is the necklace polynomial. P is a counting function. Is f also a counting function?
%D T. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, Theorem 14.8.
%D Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 500.
%H David Broadhurst, <a href="http://arxiv.org/abs/1504.05303">Multiple Landen values and the tribonacci numbers</a>, arXiv:1504.05303 [hep-th], 2015.
%F 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 Möbius function. Then f(n) = (1/n) sum_{d|n} mu(n/d) F(d) (so that n*f(n) is the Möbius inverse of F(n).)
%e f(1) = f(2) = 1 because 1 - z - z^2 - z^3 = (1-z)^1 *(1-z^2)^1 * ....
%o (Sage)
%o z = PowerSeriesRing(ZZ, 'z').gen().O(30)
%o r = (1 - (z + z**2 + z**3))
%o F = -z*r.derivative()/r
%o [sum(moebius(n//d)*F[d] for d in divisors(n))//n for n in range(1,24)]
%Y Cf. A006206, A113788.
%K nonn
%O 1,3
%A Barry Brent (barrybrent(AT)member.ams.org), Feb 04 2007