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A032192
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Number of necklaces with 7 black beads and n-7 white beads.
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10
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1, 1, 4, 12, 30, 66, 132, 246, 429, 715, 1144, 1768, 2652, 3876, 5538, 7752, 10659, 14421, 19228, 25300, 32890, 42288, 53820, 67860, 84825, 105183, 129456, 158224, 192130, 231880, 278256, 332112, 394383, 466089, 548340
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OFFSET
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7,3
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COMMENTS
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"CIK[ 7 ]" (necklace, indistinct, unlabeled, 7 parts) transform of 1, 1, 1, 1, ...
The g.f. is Z(C_7,x)/x^7, the 7-variate cycle index polynomial for the cyclic group C_7, with substitution x[i]->1/(1-x^i), i=1,...,7. Therefore by Polya enumeration a(n+7) is the number of cyclically inequivalent 7-necklaces whose 7 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_7,x) and the comment in A032191 on the equivalence of this problem with the one given in the 'Name' line. - Wolfdieter Lang, Feb 15 2005
For p prime, if a_p(n) is the number of necklaces with p black beads and n-p white beads, then (a_p(n): n>=1) = CIK[p](1, 1, 1, 1, ...). Since CIK[k](B(x)) = (1/k)*Sum_{d|k} phi(d)*B(x^d)^{k/d} with k = p and B(x) = x + x^2 + x^3 + ... = x/(1-x), we get Sum_{n>=1} a_p(n)*x^n = ((p-1)/(1 - x^p) + 1/(1 - x)^p)*x^p/p, which is Herbert Kociemba's general formula for the g.f. when p is prime.
We immediately get a_p(n) = ((p-1)/p)*I(p|n) + (1/p)*C(n-1,p-1) = ((p-1)/p)*I(p|n) + (1/n)*C(n,p) = ceiling(C(n,p)/n), which is a generalization of the conjecture made by N. J. A. Sloane and Wolfdieter Lang. (Here, I(condition) = 1 if the condition holds, and 0 otherwise. Also, as usual, for integers n and k, C(n,k) = 0 if 0 <= n < k.)
Since the sequence (a_p(n): n>=1) is column k = p of A047996(n,k) = T(n,k), we get from the documentation of the latter sequence that a_p(n) = T(n, p) = (1/n)*Sum_{d|gcd(n,p)} phi(d)*C(n/d, p/d), from which we get another proof of the formulae for a_p(n).
(End)
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LINKS
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FORMULA
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G.f.: x^7*(x^6 - 5*x^5 + 13*x^4 - 17*x^3 + 13*x^2 - 5*x + 1)/((x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(1 - x)^7). - Gael Linder (linder.gael(AT)wanadoo.fr), Jan 13 2005
G.f.: (6/(1 - x^7) + 1/(1 - x)^7)*x^7/7; in general, for a necklace with p black beads and p prime, the g.f. is ((p-1)/(1 - x^p) + 1/(1 - x)^p)*x^p/p. - Herbert Kociemba, Oct 15 2016
a(n) = (6/7)*I(7|n) + (1/7)*C(n-1,6) = (6/7)*I(7|n) + (1/n)*C(n,7), where I(condition) = 1 if the condition holds, and = 0 otherwise. - Petros Hadjicostas, Dec 08 2017
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MATHEMATICA
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k = 7; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 30}] (* Robert A. Russell, Sep 27 2004 *)
DeleteCases[CoefficientList[Series[x^7 (x^6 - 5 x^5 + 13 x^4 - 17 x^3 + 13 x^2 - 5 x + 1)/((x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) (1 - x)^7), {x, 0, 41}], x], 0] (* Michael De Vlieger, Oct 10 2016 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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