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A032193
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Number of necklaces with 8 black beads and n-8 white beads.
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5
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1, 1, 5, 15, 43, 99, 217, 429, 810, 1430, 2438, 3978, 6310, 9690, 14550, 21318, 30667, 43263, 60115, 82225, 111041, 148005, 195143, 254475, 328756, 420732, 534076, 672452, 840652, 1043460, 1287036, 1577532, 1922741
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OFFSET
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8,3
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COMMENTS
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The g.f. is Z(C_8,x)/x^8, the 8-variate cycle index polynomial for the cyclic group C_8, with substitution x[i]->1/(1-x^i), i=1,...,8. Therefore by Polya enumeration a(n+8) is the number of cyclically inequivalent 8-necklaces whose 8 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_8,x). See the comment in A032191 on the equivalence of this problem with the one given in the `Name' line. - Wolfdieter Lang, Feb 15 2005
The CIK[k] transform of sequence (c(n): n>=1) has generating function A_k(x) = (1/k)*Sum_{d|k} phi(d)*C(x^d)^{k/d}, where C(x) = Sum_{n>=1} c(n)*x^n is the g.f. of (c(n): n>=1).
When c(n) = 1 for all n >= 1, we get C(x) = x/(1-x) and A_k(x) = (x^k/k)*Sum_{d|k} phi(d)*(1-x^d)^{-k/d}, which is the g.f. of the number a_k(n) of necklaces of n beads of 2 colors with k of them black and n-k of them white.
Using Taylor expansions, we can easily prove that a_k(n) = (1/k)*Sum_{d|gcd(n,k)} phi(d)*binomial(n/d - 1, k/d - 1) = (1/n)*Sum_{d|gcd(n,k)} phi(d)*binomial(n/d, k/d), which is Robert A. Russell's formula in the Mathematica code below.
For this sequence k = 8, and thus we get the formulae below.
(End)
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LINKS
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FORMULA
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"CIK[ 8 ]" (necklace, indistinct, unlabeled, 8 parts) transform of 1, 1, 1, 1...
G.f.: (x^8)*(1-3*x+5*x^2+3*x^3-4*x^4+4*x^5+6*x^6-4*x^7+7*x^8-x^9+x^10+x^11)/((1-x)^4*(1-x^2)^2*(1-x^4)*(1-x^8)).
G.f.: 1/8*x^8*(1/(1-x)^8+1/(1-x^2)^4+2/(1-x^4)^2+4/(1-x^8)^1). - Herbert Kociemba, Oct 22 2016
a(n) = (1/8)*Sum_{d|gcd(n,8)} phi(d)*binomial(n/d - 1, 8/d - 1) = (1/n)*Sum_{d|gcd(n,8)} phi(d)*binomial(n/d, 8/d). - Petros Hadjicostas, Aug 31 2018
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MATHEMATICA
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k = 8; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 30}] (* Robert A. Russell, Sep 27 2004 *)
CoefficientList[Series[1/8*(1/(1 - x)^8 + 1/(1 - x^2)^4 + 2/(1 - x^4)^2 + 4/(1 - x^8)^1), {x, 0, 30}], x] (* Stefano Spezia, Sep 01 2018 *)
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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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