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A008613
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Molien series for 3-dimensional representation of A_5.
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1
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1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 3, 0, 4, 0, 4, 1, 5, 1, 6, 1, 7, 2, 8, 2, 9, 3, 10, 4, 11, 4, 13, 5, 14, 6, 15, 7, 17, 8, 18, 9, 20, 10, 22, 11, 23, 13, 25, 14, 27, 15, 29, 17, 31, 18, 33, 20, 35, 22, 37, 23, 40, 25, 42, 27
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OFFSET
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0,7
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COMMENTS
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Also arises in connection with Lee weight enumerators of codes over GF(5).
Partitions of n into (any number of) parts 2, 6, and 10, and at most one part 15. - Joerg Arndt, May 15 2011
The Neusel and Smith reference on Example 4 (T. Molien) on the rotation group of an icosahedron is a representation of A_5. - Michael Somos, Feb 01 2018
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REFERENCES
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D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 101.
H. Derksen and G. Kemper, Computational Invariant Theory, Springer, 2002; p. 92.
G. van der Geer, Hilbert Modular Surfaces, Springer-Verlag, 1988; p. 192.
F. Klein, Lectures on the Icosahedron ..., 2nd Rev. Ed., 1913; reprinted by Dover, NY, 1956; see pp. 236-243.
F. Klein, Werke, II, p. 354.
M. D. Neusel and L. Smith, Invariant Theory of Finite Groups, AMS, 2010, p. 55.
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (-1,1,2,1,0,0,-1,-2,-1,1,1)
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FORMULA
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G.f.: (1+x^15)/((1-x^2)*(1-x^6)*(1-x^10)) = ( -1-x+x^3+x^5+x^4-x^8-x^7 ) / ( (1+x+x^2) *(x^4+x^3+x^2+x+1) *(1+x)^2 *(x-1)^3 ).
a(n) = -a(n-1)+a(n-2)+2*a(n-3)+a(n-4)-a(n-7)-2*a(n-8)-a(n-9)+a(n-10)+a (n-11), n>10. - Harvey P. Dale, May 15 2011
a(n) = floor((n^2+3*n+105)/120+(n+1)*(-1)^n/8). - Tani Akinari, Sep 30 2014
Euler transform of length 30 sequence [0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1]. - Michael Somos, Sep 30 2014
a(n) = a(-3-n) for all n in Z.
0 = a(n) - a(n+2) - a(n+6) + a(n+8) - [mod(n, 5) == 2] for all n in Z. - Michael Somos, Sep 30 2014
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EXAMPLE
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G.f. = 1 + x^2 + x^4 + 2*x^6 + 2*x^8 + 3*x^10 + 4*x^12 + 4*x^14 + x^15 + ...
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MAPLE
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(1+x^15)/((1-x^2)*(1-x^6)*(1-x^10));
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MATHEMATICA
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CoefficientList[Series[(1+x^15)/((1-x^2)(1-x^6)(1-x^10)), {x, 0, 100}], x] (* or *) LinearRecurrence[{-1, 1, 2, 1, 0, 0, -1, -2, -1, 1, 1}, {1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 3}, 100] (* Harvey P. Dale, May 15 2011 *)
a[ n_] := Module[{m = If[ n < 0, -3 - n, n]}, m = If[ OddQ[m], m - 15, m]/2; SeriesCoefficient[ 1 / ((1 - x^1) (1 - x^3) (1 - x^5)), {x, 0, m}]]; (* Michael Somos, Feb 01 2018 *)
LinearRecurrence[{-1, 1, 2, 1, 0, 0, -1, -2, -1, 1, 1}, {1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 3}, 80] (* Harvey P. Dale, Jul 09 2019 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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