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A008614
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Molien series of 3-dimensional representation of group GL(3,2) (= L(2,7)); a simple group of order 168.
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3
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1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 2, 0, 3, 0, 3, 1, 3, 0, 4, 1, 4, 1, 5, 1, 5, 1, 6, 2, 6, 2, 7, 2, 7, 3, 8, 3, 9, 3, 9, 4, 10, 4, 11, 5, 11, 5, 12, 6, 13, 6, 14, 7, 14, 7, 16, 8, 16, 9, 17, 9, 18, 10, 19, 11, 20, 11, 21, 12, 22, 13, 23, 14, 24, 14, 25, 16, 26, 16, 28, 17, 28, 18, 30
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OFFSET
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0,13
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COMMENTS
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The simple group of order 168 expressed as a group of linear substitutions on three variables has invariants of degrees 4, 6, 14 which are rationally independent. The invariant of degree 4 is x1*x2^3 + x2*x3^3 + x3*x1^3 (Klein's quartic curve). - Michael Somos, Mar 18 2015
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REFERENCES
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D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 101.
W. Burnside, Theory of Groups of Finite Order, Dover Publications, NY, 1955, section 267, page 363. There is a typo in his formula: the term with numerator 21 should have denominator (1+x)(1-x^2). [Added by N. J. A. Sloane, Mar 01 2012]
T. A. Springer, Invariant Theory, Lecture Notes in Math., Vol. 585, Springer, p. 97.
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (-1,0,1,2,1,0,0,0,0,-1,-2,-1,0,1,1).
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FORMULA
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Euler transform of length 42 sequence [0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1]. - Michael Somos, Oct 11 2006
G.f.: (1 - x^42) / ((1 - x^4) * (1 - x^6) * (1 - x^14) * (1 - x^21)). - Michael Somos, Oct 11 2006
a(n) = a(-3 - n). a(n) = a(n-4) + a(n-6) - a(n-10) + a(n-14) - a(n-18) - a(n-20) + a(n-24) for all n in Z. - Michael Somos, Oct 11 2006
a(2*n + 21) = a(2*n) = A008671(n) for all n in Z.
a(n)= -a(n-1) +a(n-3) +2*a(n-4) +a(n-5) -a(n-10) -2*a(n-11) -a(n-12) +a(n-14) +a(n-15). - R. J. Mathar, Dec 18 2014
G.f.: (1/168) * ( 1 / (1 - x)^3 + 21 / ((1 + x) * (1 - x^2)) + 56 / (1 - x^3) + 42 / ((1 - x) * (1 + x^2)) + 24 * (1 - x) * (2 + 3*x + 2*x^2) / (1 + x + x^2 + x^3 + x^4 + x^5 + x^6)). [Burnside] - Michael Somos, Mar 18 2015
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EXAMPLE
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G.f. = 1 + x^4 + x^6 + x^8 + x^10 + 2*x^12 + 2*x^14 + 2*x^16 + 3*x^18 + ...
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MAPLE
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(1+x^21)/(1-x^4)/(1-x^6)/(1-x^14);
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MATHEMATICA
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LinearRecurrence[{-1, 0, 1, 2, 1, 0, 0, 0, 0, -1, -2, -1, 0, 1, 1}, {1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2}, 100] (* Harvey P. Dale, Jan 17 2015 *)
a[ n_] := Module[{m = If[ n < 0, -3 - n, n]}, m = If[ OddQ[m], m - 21, m] / 2; SeriesCoefficient[ 1 / ((1 - x^2) (1 - x^3) (1 - x^7)), {x, 0, m}]]; (* Michael Somos, Mar 18 2015 *)
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PROG
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(PARI) {a(n) = if( n%2, n-=21); n/=2; if( n<-11, n=-12-n); polcoeff( 1 / ((1-x^2) * (1-x^3) * (1-x^7)) + x * O(x^n), n)}; /* Michael Somos, Oct 11 2006 */
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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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