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Poincaré series [or Poincare series] (or Molien series) for a certain four-fold wreath product P_4.
5

%I #33 Sep 08 2022 08:45:12

%S 1,0,1,1,2,2,5,4,9,10,15,18,29,31,47,56,76,91,124,143,191,226,286,340,

%T 430,499,622,729,885,1035,1250,1443,1729,1997,2354,2713,3184,3635,

%U 4239,4834,5580,6344,7291,8236,9422,10619,12059,13555,15338,17153,19335,21574,24189,26921,30088,33355,37165

%N Poincaré series [or Poincare series] (or Molien series) for a certain four-fold wreath product P_4.

%D A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004, p. 202.

%H N. J. A. Sloane, <a href="/A091434/b091434.txt">Table of n, a(n) for n = 0..7000</a>

%H <a href="/index/Rec#order_42">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1,-1,0,3,-3,1,0,-1,1,1,0,-3,1,4,-2,-3,2,0,0,0, 2,-3,-2,4,1,-3,0,1,1,-1,0,1,-3,3,0,-1,-1,0,2,-1).

%F G.f.: (x^30 + x^25 + x^23 + x^22 + x^21 + 2*x^20 + x^19 + x^18 + x^17 + x^16 + 2*x^15 + x^14 + x^13 + x^12 + x^11 + 2*x^10 + x^9 + x^8 + x^7 + x^5 + 1) / ((1 - x^2)*(1 - x^3)*(1 - x^4)*(1 - x^6)^2*(1 - x^8)*(1 - x^9)*(1 - x^12)).

%p seq(coeff(series((1 +x^7 +x^8 +x^9 +x^10 +x^11 -x^24 -x^25 -x^26 -x^27 -x^28 -x^35)/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)^2*(1-x^8)*(1-x^9)*(1-x^12)), x, n+1), x, n), n = 0..70); # _G. C. Greubel_, Jan 31 2020

%t CoefficientList[Series[(1 +x^7 +x^8 +x^9 +x^10 +x^11 -x^24 -x^25 -x^26 -x^27 -x^28 -x^35)/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)^2*(1-x^8)*(1-x^9)*(1-x^12)), {x,0,70}], x] (* _G. C. Greubel_, Jan 31 2020 *)

%o (PARI) my(x='x+O('x^70)); Vec((1 +x^7 +x^8 +x^9 +x^10 +x^11 -x^24 -x^25 -x^26 -x^27 -x^28 -x^35)/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)^2*(1-x^8)*(1-x^9)*(1-x^12))) \\ _G. C. Greubel_, Jan 31 2020

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (1 +x^7 +x^8 +x^9 +x^10 +x^11 -x^24 -x^25 -x^26 -x^27 -x^28 -x^35)/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)^2*(1-x^8)*(1-x^9)*(1-x^12)) )); // _G. C. Greubel_, Jan 31 2020

%o (Sage)

%o def A091434_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P( (1 +x^7 +x^8 +x^9 +x^10 +x^11 -x^24 -x^25 -x^26 -x^27 -x^28 -x^35)/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)^2*(1-x^8)*(1-x^9)*(1-x^12)) ).list()

%o A091434_list(70) # _G. C. Greubel_, Jan 31 2020

%Y Cf. A082146, A089599, A091726, A091769.

%K nonn

%O 0,5

%A _N. J. A. Sloane_, Mar 17 2004

%E G.f. and data corrected by _N. J. A. Sloane_, Jan 05 2017