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%I #12 May 01 2020 15:53:18
%S 1,0,0,0,0,0,1,0,2,1,1,1,3,1,3,3,4,3,5,3,7,6,7,8,12,8,12,12,14,14,18,
%T 15,22,20,22,24,30,26,33,33,37,37,43,40,50,49,52,55,64,59,69,70,76,78,
%U 86,83,97,96,101,106,118,113,126,128,137,140,151,149,167,167,174,181,197,192
%N Poincaré series [or Poincare series] (or Molien series) for GL_2(4):2.
%D A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 269.
%H <a href="/index/Rec#order_36">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 1, -1, 0, 1, -1, 0, 0, 0, -1, 1, 0, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1).
%F G.f. = p(x)/q(x), where
%F p(x) = 1 + x^6 + 2*x^8 + x^9 + x^11 + 2*x^12 + x^13 + 3*x^14 + 2*x^15 + 3*x^16 + 3*x^17 + 2*x^18 + 2*x^19 + 4*x^20 + 3*x^21 + 4*x^22 + 4*x^23 + 4*x^24 + 4*x^25 + 4*x^26 + 4*x^27 + 5*x^28 + 5*x^29 + 4*x^30 + 4*x^31 + 4*x^32 + 4*x^33 + 4*x^34 + 4*x^35 + 3*x^36 + 4*x^37 + 2*x^38 + 2*x^39 + 3*x^40 + 3*x^41 + 2*x^42 + 3*x^43 + x^44 + 2*x^45 + x^46 + x^48 + 2*x^49 + x^51 + x^57,
%F and q(x) = (1-x^10)*(1-x^12)*(1-x^15)*(1-x^24).
%F a(0)=1, a(1)=0, a(2)=0, a(3)=0, a(4)=0, a(5)=0, a(6)=1, a(7)=0, a(8)=2, a(9)=1, a(10)=1, a(11)=1, a(12)=3, a(13)=1, a(14)=3, a(15)=3, a(16)=4, a(17)=3, a(18)=5, a(19)=3, a(20)=7, a(21)=6, a(22)=7, a(23)=8, a(24)=12, a(25)=8, a(26)=12, a(27)=12, a(28)=14, a(29)=14, a(30)=18, a(31)=15, a(32)=22, a(33)=20, a(34)=22, a(35)=24, a(n)=a(n-1)+a(n-8)-a(n-9)+ a(n-12)- a(n-13)+a(n-15)-a(n-16)-a(n-20)+a(n-21)-a(n-23)+a(n-24)-a(n-27)+ a(n-28)+ a(n-35)- a(n-36). - _Harvey P. Dale_, Oct 21 2013
%t With[{num=1+x^6+2x^8+x^9+x^11+2x^12+x^13+3x^14+ 2x^15+3x^16+3x^17+ 2x^18+ 2x^19+4x^20+ 3x^21+ 4x^22+4x^23+4x^24+4x^25+4x^26+4x^27+5x^28+ 5x^29+ 4x^30+ 4x^31+ 4x^32+4x^33+4x^34+ 4x^35+ 3x^36+4x^37+2x^38+2x^39+ 3x^40+ 3x^41+2x^42+ 3x^43+x^44+ 2x^45+x^46+ x^48+2x^49+ x^51+x^57,den=(1-x^10)(1-x^12)(1-x^15)(1-x^24)},CoefficientList[Series[num/den,{x,0,80}],x]] (* or *) LinearRecurrence[ {1,0,0,0,0,0,0,1,-1,0,0,1,-1,0,1,-1,0,0,0,-1,1,0,-1,1,0,0,-1,1,0,0,0,0,0,0,1,-1},{1,0,0,0,0,0,1,0,2,1,1,1,3,1,3,3,4,3,5,3,7,6,7,8,12,8,12,12,14,14,18,15,22,20,22,24},80](* _Harvey P. Dale_, Oct 21 2013 *)
%K nonn,easy
%O 0,9
%A _N. J. A. Sloane_, Mar 18 2004
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