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A091981
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Poincaré series [or Poincare series] (or Molien series) for GL_2(4):2.
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0
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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, 30, 26, 33, 33, 37, 37, 43, 40, 50, 49, 52, 55, 64, 59, 69, 70, 76, 78, 86, 83, 97, 96, 101, 106, 118, 113, 126, 128, 137, 140, 151, 149, 167, 167, 174, 181, 197, 192
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,9
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REFERENCES
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A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 269.
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LINKS
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Index entries for linear recurrences with constant coefficients, 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).
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FORMULA
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G.f. = p(x)/q(x), where
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,
and q(x) = (1-x^10)*(1-x^12)*(1-x^15)*(1-x^24).
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
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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