cp's OEIS Frontend

A008610 Molien series of 4-dimensional representation of cyclic group of order 4 over GF(2) (not Cohen-Macaulay).

%I A008610 #68 Mar 04 2025 07:55:39
%S A008610 1,1,3,5,10,14,22,30,43,55,73,91,116,140,172,204,245,285,335,385,446,
%T A008610 506,578,650,735,819,917,1015,1128,1240,1368,1496,1641,1785,1947,2109,
%U A008610 2290,2470,2670,2870,3091,3311,3553,3795,4060,4324,4612,4900,5213,5525,5863
%N A008610 Molien series of 4-dimensional representation of cyclic group of order 4 over GF(2) (not Cohen-Macaulay).
%C A008610 a(n) is the number of necklaces with 4 black beads and n white beads.
%C A008610 Also nonnegative integer 2 X 2 matrices with sum of elements equal to n, up to rotational symmetry.
%C A008610 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 A008610 D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 104.
%D A008610 E. V. McLaughlin, Numbers of factorizations in non-unique factorial domains, Senior Thesis, Allegeny College, Meadville, PA, April 2004.
%H A008610 G. C. Greubel, <a href="/A008610/b008610.txt">Table of n, a(n) for n = 0..1000</a>
%H A008610 David Broadhurst and Xavier Roulleau, <a href="https://arxiv.org/abs/2502.19523">Number of partitions of modular integers</a>, arXiv:2502.19523 [math.NT], 2025. See p. 19.
%H A008610 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 A008610 <a href="/index/Gre#groups">Index entries for sequences related to groups</a>
%H A008610 <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H A008610 <a href="/index/Ne#necklaces">Index entries for sequences related to necklaces</a>
%H A008610 <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 A008610 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 A008610 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 A008610 G.f.: (1/4)*(1/(1-x)^4 + 1/(1-x^2)^2 + 2/(1-x^4)). - _Herbert Kociemba_, Oct 22 2016
%F A008610 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 A008610 There are 10 inequivalent nonnegative integer 2 X 2 matrices with sum of elements equal to 4, up to rotational symmetry:
%e A008610 [0 0] [0 0] [0 0] [0 0] [0 1] [0 1] [0 1] [0 2] [0 2] [1 1]
%e A008610 [0 4] [1 3] [2 2] [3 1] [1 2] [2 1] [3 0] [1 1] [2 0] [1 1].
%p A008610 1/(1-x)/(1-x^2)^2/(1-x^4)*(1+2*x^3+x^4);
%p A008610 seq(coeff(series(%, x, n+1), x, n), n=0..40);
%t A008610 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 A008610 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 A008610 (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 A008610 (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 A008610 (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 A008610 (Sage)
%o A008610 def A008610_list(prec):
%o A008610     P.<x> = PowerSeriesRing(ZZ, prec)
%o A008610     return P( (1+2*x^3+x^4)/((1-x)*(1-x^2)^2*(1-x^4)) ).list()
%o A008610 A008610_list(50) # _G. C. Greubel_, Jan 31 2020
%o A008610 (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 A008610 Row n=2 of A343874.
%Y A008610 Column k=4 of A037306 and A047996.
%Y A008610 Cf. A000031, A005232, A008804, A047996, A032801.
%K A008610 nonn,easy
%O A008610 0,3
%A A008610 _N. J. A. Sloane_
%E A008610 Comment and example from _Vladeta Jovovic_, May 18 2000