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 A008646 Molien series for cyclic group of order 5. 17
 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 (list; graph; refs; listen; history; text; internal format)
 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 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: A193911 A206417 A207381 * A036830 A014153 A001924 Adjacent sequences:  A008643 A008644 A008645 * A008647 A008648 A008649 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified August 10 19:52 EDT 2020. Contains 336381 sequences. (Running on oeis4.)