

A008610


Molien series of 4dimensional representation of cyclic group of order 4 over GF(2) (not CohenMacaulay).


12



1, 1, 3, 5, 10, 14, 22, 30, 43, 55, 73, 91, 116, 140, 172, 204, 245, 285, 335, 385, 446, 506, 578, 650, 735, 819, 917, 1015, 1128, 1240, 1368, 1496, 1641, 1785, 1947, 2109, 2290, 2470, 2670, 2870, 3091
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OFFSET

0,3


COMMENTS

a(n) = number of necklaces with 4 black beads and n white beads.
Also nonnegative integer 2 X 2 matrices with sum of elements equal to n, up to rotational symmetry.
The g.f. is Z(C_4,x), the 4variate cycle index polynomial for the cyclic group C_4, with substitution x[i]>1/(1x^i), i=1,...,4. Therefore by Polya enumeration a(n) is the number of cyclically inequivalent 4necklaces 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.


REFERENCES

D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 104.
E. V. McLaughlin, Numbers of factorizations in nonunique factorial domains, Senior Thesis, Allegeny College, Meadville, PA, April 2004.


LINKS

Table of n, a(n) for n=0..40.
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 (2,0,2,2,2,0,2,1).


FORMULA

G.f.: (1+2*x^3+x^4)/((1x)*(1x^2)^2*(1x^4)) = (1x+x^2+x^3)/((1x)^2*(1x^2)*(1x^4)).
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
G.f.: (1/4)*(1/(1x)^4+1/(1x^2)^2+2/(1x^4)).  Herbert Kociemba, Oct 22 2016


EXAMPLE

There are 10 inequivalent nonnegative integer 2 X 2 matrices with sum of elements equal to 4, up to rotational symmetry:
[0 0] [0 0] [0 0] [0 0] [0 1] [0 1] [0 1] [0 2] [0 2] [1 1]
[0 4] [1 3] [2 2] [3 1] [1 2] [2 1] [3 0] [1 1] [2 0] [1 1].


MAPLE

1/(1x)/(1x^2)^2/(1x^4)*(1+2*x^3+x^4);


MATHEMATICA

k = 4; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 30}] (* Robert A. Russell, Sep 27 2004 *)


PROG

(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


CROSSREFS

Cf. A000031, A047996, A005232, A008804.
Sequence in context: A001841 A266793 A176222 * A281688 A078411 A137630
Adjacent sequences: A008607 A008608 A008609 * A008611 A008612 A008613


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane


EXTENSIONS

Comment and example from Vladeta Jovovic, May 18 2000


STATUS

approved



