A091726 Poincaré series [or Poincare series] (or Molien series) for a certain five-fold wreath product P_5.

1, 0, 1, 1, 2, 2, 5, 4, 9, 10, 16, 19, 31, 34, 53, 64, 89, 109, 152, 180, 246, 298, 388, 473, 612, 732, 935, 1125, 1402, 1685, 2086, 2478, 3041, 3610, 4366, 5169, 6213, 7295, 8712, 10202, 12068, 14083, 16571, 19221, 22500, 26014, 30244, 34850, 40338, 46256, 53313

Offset: 0

Author

N. J. A. Sloane, Mar 17 2004

Keywords

• nonn
• easy

References

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

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,0,-1,-1,0,3,-3,1,0,0,-1,1,1,-2,2,-1,1,-3,3,1,-4,2,1,0,-3,1,2,-1,0,0,0,-2,2,2, -2,0,0,0,-1,2,1,-3,0,1,2,-4,1,3,-3,1,-1,2,-2,1,1,-1,0,0,1,-3,3,0,-1,-1,0,2,-1).

Crossrefs

Cf. A082146, A089599, A091434, A091769.

Programs

Mathematica

CoefficientList[Series[(1 -2*x +x^2 +x^4 -x^5 +x^8 -x^9 +x^10 -x^13 +2*x^14 -x^15 +x^16 -x^17 +2*x^18 -2*x^19 +2*x^20 -x^21 +2*x^22 -2*x^23 + 2*x^24 -x^25 +x^26 -x^27 +2*x^28 -x^29 +x^32 -x^33 +x^34 -x^37 +x^38 +x^40 -2*x^41 +x^42)/( (1-x)^2*(1-x^2)*(1-x^3)*(1-x^6)*(1-x^8)*(1-x^9)*(1-x^12)*(1-x^10)*(1-x^15)), {x,0,60}], x] (* G. C. Greubel, Jan 31 2020 *)

PARI

Vec((x^50 +x^45 +x^43 +x^42 +x^41 +2*x^40 +2*x^39 +2*x^38 +2*x^37 +3*x^36 +3*x^35 +3*x^34 +3*x^33 +4*x^32 +4*x^31 +4*x^30 +5*x^29 +5*x^28 +5*x^27 +5*x^26 +6*x^25 +5*x^24 +5*x^23 +5*x^22 +5*x^21 +4*x^20 +4*x^19 +4*x^18 +3*x^17 +3*x^16 +3*x^15 +3*x^14 +2*x^13 +2*x^12 +2*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)*(1 -x^10)*(1 -x^15)) + O(x^60)) \\ Colin Barker, Mar 15 2015

Formula

G.f.: ( x^50 + x^45 + x^43 + x^42 + x^41 + 2*x^40 + 2*x^39 + 2*x^38 + 2*x^37 + 3*x^36 + 3*x^35 + 3*x^34 + 3*x^33 + 4*x^32 + 4*x^31 + 4*x^30 + 5*x^29 + 5*x^28 + 5*x^27 + 5*x^26 + 6*x^25 + 5*x^24 + 5*x^23 + 5*x^22 + 5*x^21 + 4*x^20 + 4*x^19 + 4*x^18 + 3*x^17 + 3*x^16 + 3*x^15 + 3*x^14 + 2*x^13 + 2*x^12 + 2*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)*(1 - x^10)*(1 - x^15)).

Extensions

Three typos (unbalanced parentheses) in g.f. fixed by Colin Barker, Mar 15 2015