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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.

%I A008614 #45 Dec 16 2021 22:33:45
%S A008614 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,
%T A008614 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,
%U A008614 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
%N A008614 Molien series of 3-dimensional representation of group GL(3,2) (= L(2,7)); a simple group of order 168.
%C A008614 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
%D A008614 D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 101.
%D A008614 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]
%D A008614 T. A. Springer, Invariant Theory, Lecture Notes in Math., Vol. 585, Springer, p. 97.
%H A008614 T. D. Noe, <a href="/A008614/b008614.txt">Table of n, a(n) for n = 0..1000</a>
%H A008614 <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H A008614 <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (-1,0,1,2,1,0,0,0,0,-1,-2,-1,0,1,1).
%F A008614 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
%F A008614 G.f.: (1 - x^42) / ((1 - x^4) * (1 - x^6) * (1 - x^14) * (1 - x^21)). - _Michael Somos_, Oct 11 2006
%F A008614 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
%F A008614 a(2*n + 21) = a(2*n) = A008671(n) for all n in Z.
%F A008614 a(n) ~ 1/336*n^2. - _Ralf Stephan_, Apr 29 2014
%F A008614 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
%F A008614 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
%e A008614 G.f. = 1 + x^4 + x^6 + x^8 + x^10 + 2*x^12 + 2*x^14 + 2*x^16 + 3*x^18 + ...
%p A008614 (1+x^21)/(1-x^4)/(1-x^6)/(1-x^14);
%t A008614 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 *)
%t A008614 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 *)
%o A008614 (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 */
%Y A008614 Cf. A008671.
%K A008614 nonn,easy,nice
%O A008614 0,13
%A A008614 _N. J. A. Sloane_