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%I #15 May 01 2020 15:51:50
%S 1,0,1,3,3,4,7,8,9,12,14,17,20,22,26,30,32,36,42,45,49,55,59,64,70,75,
%T 81,88,93,99,107,113,120,128,135,143,151,158,167,177,184,193,204,212,
%U 221,232,242,252,263,273,284,296,306,318,331,342,354,367,379,392,406,418,432,447
%N Poincaré series [or Poincare series] (or Molien series) for a certain three-dimensional group of order 1344.
%D A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 259.
%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (1,-1,2,-1,1,-1,1,-1,1,-2,1,-1,1).
%F G.f.: (1 + x^2 + 3*x^3 + 2*x^4 + 4*x^5 + 5*x^6 + 4*x^7 + 5*x^8 + 4*x^9 + 2*x^10 + 3*x^11 + x^12 + x^14 ) / ( (1-x^4)*(1-x^6)*(1-x^7)).
%F G.f.: ( -1-2*x^2-x^5-x^10+x-2*x^4-2*x^6-2*x^8+x^9 ) / ( (1+x+x^2) *(x^6+x^5+x^4+x^3+x^2+x+1) *(1+x^2) *(x-1)^3 ). - _R. J. Mathar_, Dec 18 2014
%t LinearRecurrence[{1,-1,2,-1,1,-1,1,-1,1,-2,1,-1,1},{1,0,1,3,3,4,7,8,9,12,14,17,20},64]
%t (* _Ray Chandler_, Jul 15 2015 *)
%o (PARI) a(n)=([0,1,0,0,0,0,0,0,0,0,0,0,0; 0,0,1,0,0,0,0,0,0,0,0,0,0; 0,0,0,1,0,0,0,0,0,0,0,0,0; 0,0,0,0,1,0,0,0,0,0,0,0,0; 0,0,0,0,0,1,0,0,0,0,0,0,0; 0,0,0,0,0,0,1,0,0,0,0,0,0; 0,0,0,0,0,0,0,1,0,0,0,0,0; 0,0,0,0,0,0,0,0,1,0,0,0,0; 0,0,0,0,0,0,0,0,0,1,0,0,0; 0,0,0,0,0,0,0,0,0,0,1,0,0; 0,0,0,0,0,0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,0,0,0,0,0,1; 1,-1,1,-2,1,-1,1,-1,1,-1,2,-1,1]^n*[1;0;1;3;3;4;7;8;9;12;14;17;20])[1,1] \\ _Charles R Greathouse IV_, May 01 2020
%K nonn,easy
%O 0,4
%A _N. J. A. Sloane_, Mar 18 2004
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