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 A008614 Molien series of 3-dimensional representation of group GL(3,2) (= L(2,7)); a simple group of order 168. 3
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,13 COMMENTS 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 REFERENCES 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. LINKS T. D. Noe, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (-1,0,1,2,1,0,0,0,0,-1,-2,-1,0,1,1). FORMULA 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) ~ 1/336*n^2. - Ralf Stephan, Apr 29 2014 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 EXAMPLE G.f. = 1 + x^4 + x^6 + x^8 + x^10 + 2*x^12 + 2*x^14 + 2*x^16 + 3*x^18 + ... MAPLE (1+x^21)/(1-x^4)/(1-x^6)/(1-x^14); MATHEMATICA 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 *) PROG (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 */ CROSSREFS Cf. A008671. Sequence in context: A281009 A284443 A260160 * A036663 A209457 A096577 Adjacent sequences:  A008611 A008612 A008613 * A008615 A008616 A008617 KEYWORD nonn,easy,nice AUTHOR STATUS approved

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Last modified December 12 01:57 EST 2019. Contains 329948 sequences. (Running on oeis4.)