

A032192


Number of necklaces with 7 black beads and n7 white beads.


6



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

7,3


COMMENTS

"CIK[ 7 ]" (necklace, indistinct, unlabeled, 7 parts) transform of 1, 1, 1, 1, ...
The g.f. is Z(C_7,x)/x^7, the 7variate cycle index polynomial for the cyclic group C_7, with substitution x[i]>1/(1x^i), i=1,...,7. Therefore by Polya enumeration a(n+7) is the number of cyclically inequivalent 7necklaces 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
From Petros Hadjicostas, Dec 08 2017: (Start)
For p prime, if a_p(n) is the number of necklaces with p black beads and np white beads, then (a_p(n): n>=1) = CIK[p](1, 1, 1, 1, ...). Since CIK[k](B(x)) = (1/k)*Sum_{dk} phi(d)*B(x^d)^{k/d} with k = p and B(x) = x + x^2 + x^3 + ... = x/(1x), we get Sum_{n>=1} a_p(n)*x^n = ((p1)/(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) = ((p1)/p)*I(pn) + (1/p)*C(n1,p1) = ((p1)/p)*I(pn) + (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_{dgcd(n,p)} phi(d)*C(n/d, p/d), from which we get another proof of the formulae for a_p(n).
(End)


LINKS

Table of n, a(n) for n=7..41.
C. G. Bower, Transforms (2)
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
Index entries for sequences related to necklaces


FORMULA

Empirically this is ceiling(C(n, 7)/n).  N. J. A. Sloane
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 ((p1)/(1  x^p) + 1/(1  x)^p)*x^p/p.  Herbert Kociemba, Oct 15 2016
a(n) = ceiling(binomial(n, 7)/n)) (conjecture by Wolfdieter Lang).
a(n) = (6/7)*I(7n) + (1/7)*C(n1,6) = (6/7)*I(7n) + (1/n)*C(n,7), where I(condition) = 1 if the condition holds, and = 0 otherwise.  Petros Hadjicostas, Dec 08 2017


MATHEMATICA

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 *)


CROSSREFS

Column k=7 of A047996.
Cf. A004526, A007997, A008610, A008646, A032191, A053618.
Sequence in context: A004036 A011797 A051172 * A212587 A118425 A097809
Adjacent sequences: A032189 A032190 A032191 * A032193 A032194 A032195


KEYWORD

nonn


AUTHOR

Christian G. Bower


STATUS

approved



