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A008610
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Molien series of 4-dimensional representation of cyclic group of order 4 over GF(2) (not Cohen-Macaulay).
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10
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| a(n-4)=number of necklaces with 4 black beads and n-4 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 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). W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Feb 15 2005.
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REFERENCES
| D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 104.
E. V. McLaughlin, Numbers of factorizations in non-unique factorial domains, Senior Thesis, Allegeny College, Meadville, PA, April 2004.
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LINKS
| Index entries for Molien series
Index entries for sequences related to necklaces
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FORMULA
| 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)).
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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].
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MAPLE
| 1/(1-x)/(1-x^2)^2/(1-x^4)*(1+2*x^3+x^4);
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MATHEMATICA
| k = 4; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 30}] - Robert A. Russell (russell(AT)post.harvard.edu), Sep 27 2004
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CROSSREFS
| Cf. A000031, A047996, A005232, A008804.
Sequence in context: A195094 A001841 A176222 * A078411 A137630 A092269
Adjacent sequences: A008607 A008608 A008609 * A008611 A008612 A008613
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Comment and example from Vladeta Jovovic (vladeta(AT)eunet.rs), May 18 2000
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