A091434 Poincaré series [or Poincare series] (or Molien series) for a certain four-fold wreath product P_4.

1, 0, 1, 1, 2, 2, 5, 4, 9, 10, 15, 18, 29, 31, 47, 56, 76, 91, 124, 143, 191, 226, 286, 340, 430, 499, 622, 729, 885, 1035, 1250, 1443, 1729, 1997, 2354, 2713, 3184, 3635, 4239, 4834, 5580, 6344, 7291, 8236, 9422, 10619, 12059, 13555, 15338, 17153, 19335, 21574, 24189, 26921, 30088, 33355, 37165

Offset: 0

Author

N. J. A. Sloane, Mar 17 2004

Keywords

• nonn

References

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

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..7000
• Index entries for linear recurrences with constant coefficients, 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).

Crossrefs

Cf. A082146, A089599, A091726, A091769.

Programs

Magma

R:=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

Maple

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

Mathematica

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 *)

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

Sage

def A091434_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    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()
A091434_list(70) # G. C. Greubel, Jan 31 2020

Formula

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)).

Extensions

G.f. and data corrected by N. J. A. Sloane, Jan 05 2017