A008646 Molien series for cyclic group of order 5.

1, 1, 3, 7, 14, 26, 42, 66, 99, 143, 201, 273, 364, 476, 612, 776, 969, 1197, 1463, 1771, 2126, 2530, 2990, 3510, 4095, 4751, 5481, 6293, 7192, 8184, 9276, 10472, 11781, 13209, 14763, 16451, 18278, 20254, 22386, 24682, 27151, 29799, 32637, 35673

Offset

0,3

Comments

a(n) is the number of necklaces with 5 black beads and n white beads.

The g.f. is Z(C_5,x), the 5-variate cycle index polynomial for the cyclic group C_5, with substitution x[i]->1/(1-x^i), i=1,...,5. Therefore by Polya enumeration a(n) is the number of cyclically inequivalent 5-necklaces whose 5 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_5,x). - Wolfdieter Lang, Feb 15 2005

References

B. Sturmfels, Algorithms in Invariant Theory, Springer, '93, p. 65.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Mónica A. Reyes, Cristina Dalfó, Miguel Àngel Fiol, and Arnau Messegué, A general method to find the spectrum and eigenspaces of the k-token of a cycle, and 2-token through continuous fractions, arXiv:2403.20148 [math.CO], 2024. See p. 6.

Index entries for sequences related to groups

Index entries for Molien series

Index entries for sequences related to necklaces

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1,1,-4,6,-4,1).

Formula

G.f.: (1 +x^2 +3*x^3 +4*x^4 +6*x^5 +4*x^6 +3*x^7 +x^8 +x^10)/((1-x)*(1-x^2)*(1-x^3)*(1- x^4)*(1-x^5)).

a(-5-n) = a(n) for all integers.

a(n) = ceiling( binomial(n+5, 5) / (n+5) ).

G.f.: (1 -3*x +5*x^2 -3*x^3 +x^4)/((1-x)^4*(1-x^5)). - Michael Somos, Dec 04, 2001

a(n) = (n^4 +10*n^3 +35*n^2 +50*n +24*(3 -2*(-1)^(2^(n-5*floor(n/5)) )))/120. - Luce ETIENNE, Oct 31 2015

G.f.: (4/(1-x^5) + 1/(1-x)^5)/5. - Herbert Kociemba, Oct 15 2016

Maple

seq(coeff(series((1+x^2+3*x^3+4*x^4+6*x^5+4*x^6+3*x^7+x^8+x^10)/((1-x)* (1-x^2)*(1-x^3)*(1- x^4)*(1-x^5)), x, n+1), x, n), n = 0..50); # corrected by G. C. Greubel, Sep 06 2019
seq(ceil(binomial(n, 4)/5), n=4..41); # Zerinvary Lajos, Jan 12 2009

Mathematica

k = 5; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 50}] (* Robert A. Russell, Sep 27 2004 *)
CoefficientList[Series[(1 +x^2 +3*x^3 +4*x^4 +6*x^5 +4*x^6 +3*x^7 +x^8 +x^10)/((1-x)*(1-x^2)*(1-x^3)*(1- x^4)*(1-x^5)), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 11 2013 *)
LinearRecurrence[{4, -6, 4, -1, 1, -4, 6, -4, 1}, {1, 1, 3, 7, 14, 26, 42, 66, 99}, 50] (* Harvey P. Dale, Jan 11 2017 *)

Prog

(PARI) a(n)=ceil((n+4)*(n+3)*(n+2)*(n+1)/120)
(Magma) [Ceiling((n+4)*(n+3)*(n+2)*(n+1)/120): n in [0..50]]; // Vincenzo Librandi, Jun 11 2013
(PARI) Vec((1-3*x+5*x^2-3*x^3+x^4)/((1-x)^4*(1-x^5)) + O(x^50)) \\ Altug Alkan, Oct 31 2015
(Sage) [ceil(binomial(n+5, 5)/(n+5)) for n in (0..50)] # G. C. Greubel, Sep 06 2019

Crossrefs

Cf. A000031, A047996.

Sequence in context: A206417 A207381 A343017 * A036830 A014153 A001924

Adjacent sequences: A008643 A008644 A008645 * A008647 A008648 A008649

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved