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A008646 Molien series for cyclic group of order 5. 17

%I

%S 1,1,3,7,14,26,42,66,99,143,201,273,364,476,612,776,969,1197,1463,

%T 1771,2126,2530,2990,3510,4095,4751,5481,6293,7192,8184,9276,10472,

%U 11781,13209,14763,16451,18278,20254,22386,24682,27151,29799,32637,35673

%N Molien series for cyclic group of order 5.

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

%C 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

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

%H Vincenzo Librandi, <a href="/A008646/b008646.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>

%H <a href="/index/Mo#Molien">Index entries for Molien series</a>

%H <a href="/index/Ne#necklaces">Index entries for sequences related to necklaces</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1,1,-4,6,-4,1).

%F 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)).

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

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

%F G.f.: (1 -3*x +5*x^2 -3*x^3 +x^4)/((1-x)^4*(1-x^5)). - _Michael Somos_, Dec 04, 2001

%F 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

%F G.f.: (4/(1-x^5) + 1/(1-x)^5)/5. - _Herbert Kociemba_, Oct 15 2016

%p 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

%p seq(ceil(binomial(n,4)/5), n=4..41); # _Zerinvary Lajos_, Jan 12 2009

%t k = 5; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 50}] (* _Robert A. Russell_, Sep 27 2004 *)

%t 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 *)

%t 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 *)

%o (PARI) a(n)=ceil((n+4)*(n+3)*(n+2)*(n+1)/120)

%o (MAGMA) [Ceiling((n+4)*(n+3)*(n+2)*(n+1)/120): n in [0..50]]; // _Vincenzo Librandi_, Jun 11 2013

%o (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

%o (Sage) [ceil(binomial(n+5,5)/(n+5)) for n in (0..50)] # _G. C. Greubel_, Sep 06 2019

%Y Cf. A000031, A047996.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_

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Last modified July 31 06:15 EDT 2021. Contains 346369 sequences. (Running on oeis4.)