%I #65 Apr 04 2024 10:12:50
%S 1,1,3,5,10,14,22,30,43,55,73,91,116,140,172,204,245,285,335,385,446,
%T 506,578,650,735,819,917,1015,1128,1240,1368,1496,1641,1785,1947,2109,
%U 2290,2470,2670,2870,3091,3311,3553,3795,4060,4324,4612,4900,5213,5525,5863
%N Molien series of 4-dimensional representation of cyclic group of order 4 over GF(2) (not Cohen-Macaulay).
%C a(n) is the number of necklaces with 4 black beads and n white beads.
%C Also nonnegative integer 2 X 2 matrices with sum of elements equal to n, up to rotational symmetry.
%C The g.f. is Z(C_4,x), the 4-variate cycle index polynomial for the cyclic group C_4, with substitution x[i]->1/(1-x^i), i=1,...,4. Therefore by Polya enumeration a(n) is the number of cyclically inequivalent 4-necklaces whose 4 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_4,x). - _Wolfdieter Lang_, Feb 15 2005
%D D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 104.
%D E. V. McLaughlin, Numbers of factorizations in non-unique factorial domains, Senior Thesis, Allegeny College, Meadville, PA, April 2004.
%H G. C. Greubel, <a href="/A008610/b008610.txt">Table of n, a(n) for n = 0..1000</a>
%H Mónica A. Reyes, Cristina Dalfó, Miguel Àngel Fiol, and Arnau Messegué, <a href="https://arxiv.org/abs/2403.20148">A general method to find the spectrum and eigenspaces of the k-token of a cycle, and 2-token through continuous fractions</a>, arXiv:2403.20148 [math.CO], 2024. See p. 6.
%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_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,2,-2,0,2,-1).
%F G.f.: (1+2*x^3+x^4)/((1-x)*(1-x^2)^2*(1-x^4)) = (1-x+x^2+x^3)/((1-x)^2*(1-x^2)*(1-x^4)).
%F a(n) = (1/48)*(2*n^3 + 3*(-1)^n*(n + 4) + 12*n^2 + 25*n + 24 + 12*cos(n*Pi/2)). - _Ralf Stephan_, Apr 29 2014
%F G.f.: (1/4)*(1/(1-x)^4 + 1/(1-x^2)^2 + 2/(1-x^4)). - _Herbert Kociemba_, Oct 22 2016
%F a(n) = -A032801(-n), per formulae of Colin Barker (A032801) and R. Stephan (above). Also, a(n) - A032801(n+4) = (1+(-1)^signum(n mod 4))/2, i.e., (1,0,0,0,1,0,0,0,...) repeating, (offset 0). - _Gregory Gerard Wojnar_, Jul 09 2022
%e There are 10 inequivalent nonnegative integer 2 X 2 matrices with sum of elements equal to 4, up to rotational symmetry:
%e [0 0] [0 0] [0 0] [0 0] [0 1] [0 1] [0 1] [0 2] [0 2] [1 1]
%e [0 4] [1 3] [2 2] [3 1] [1 2] [2 1] [3 0] [1 1] [2 0] [1 1].
%p 1/(1-x)/(1-x^2)^2/(1-x^4)*(1+2*x^3+x^4);
%p seq(coeff(series(%, x, n+1), x, n), n=0..40);
%t k = 4; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 30}] (* _Robert A. Russell_, Sep 27 2004 *)
%t LinearRecurrence[{2,0,-2,2,-2,0,2,-1}, {1,1,3,5,10,14,22,30}, 50] (* _G. C. Greubel_, Jan 31 2020 *)
%o (PARI) a(n)=if(n,([0,1,0,0,0,0,0,0; 0,0,1,0,0,0,0,0; 0,0,0,1,0,0,0,0; 0,0,0,0,1,0,0,0; 0,0,0,0,0,1,0,0; 0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,1; -1,2,0,-2,2,-2,0,2]^n*[1;1;3;5;10;14;22;30])[1,1],1) \\ _Charles R Greathouse IV_, Oct 22 2015
%o (PARI) my(x='x+O('x^50)); Vec((1+2*x^3+x^4)/((1-x)*(1-x^2)^2*(1-x^4))) \\ _G. C. Greubel_, Jan 31 2020
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1+2*x^3+x^4)/((1-x)*(1-x^2)^2*(1-x^4)) )); // _G. C. Greubel_, Jan 31 2020
%o (Sage)
%o def A008610_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( (1+2*x^3+x^4)/((1-x)*(1-x^2)^2*(1-x^4)) ).list()
%o A008610_list(50) # _G. C. Greubel_, Jan 31 2020
%o (GAP) a:=[1,1,3,5,10,14,22,30];; for n in [9..50] do a[n]:=2*a[n-1]-2*a[n-3] +2*a[n-4]-2*a[n-5]+2*a[n-7]-a[n-1]; od; a; # _G. C. Greubel_, Jan 31 2020
%Y Row n=2 of A343874.
%Y Column k=4 of A037306 and A047996.
%Y Cf. A000031, A005232, A008804, A047996, A032801.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
%E Comment and example from _Vladeta Jovovic_, May 18 2000