A082146 -id:A082146 - cp's OEIS frontend

A089599 G.f.: (1+x^5+x^7+x^8+x^10+x^15)/((1-x^2)(1-x^3)(1-x^4)(1-x^6)^2(1-x^9)).

1, 0, 1, 1, 2, 2, 5, 4, 8, 9, 13, 15, 23, 24, 35, 40, 52, 60, 79, 87, 112, 127, 155, 177, 216, 240, 290, 326, 382, 430, 503, 557, 648, 720, 822, 914, 1041, 1144, 1298, 1428, 1600, 1760, 1967, 2146, 2392, 2609, 2882, 3142, 3463, 3752, 4127, 4468, 4882, 5282, 5760, 6202

Offset: 0

N. J. A. Sloane, Dec 31 2003

nonn

Poincaré series [or Poincare series] (or Molien series) for (P[x_0, x_1] ⊗ P[x_0, x_1] ⊗ P[x_0, x_1] )^(S_3).

A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 200.

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, -2, 1, -1, 0, 2, -2, 1, 0, 0, 1, -2, 2, 0, -1, 1, -2, 0, 0, 1, 1, -1).

Cf. A082146, A091434, A091726, A091769.

Vec((1+x^5+x^7+x^8+x^10+x^15)/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^6)^2*(1-x^9)) + O(x^100)) \\ Michel Marcus, Mar 19 2014

G.f.: (1-x+x^5-x^9+x^10)/((1-x)^2*(1-x^2)*(1-x^4)*(1-x^6)*(1-x^9)*(1+x+x^2)). See also the NAME. - Wolfdieter Lang, Mar 19 2014

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

Original entry on oeis.org

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

N. J. A. Sloane, Mar 17 2004

nonn

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

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).

Cf. A082146, A089599, A091726, A091769.

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

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

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 *)

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

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

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)).

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

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

Original entry on oeis.org

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

N. J. A. Sloane, Mar 17 2004

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

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).

Cf. A082146, A089599, A091434, A091769.

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 *)

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

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)).

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

A091769 Poincaré series [or Poincare series] (or Molien series) for a certain six-fold wreath product P_6.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 5, 4, 9, 10, 16, 19, 32, 35, 55, 67, 95, 117, 166, 199, 276, 339, 449, 555, 731, 889, 1154, 1413, 1794, 2193, 2764, 3347, 4181, 5058, 6233, 7519, 9208, 11027, 13411, 16015, 19307, 22970, 27538, 32582, 38851, 45805, 54265, 63747, 75170, 87896, 103179

Offset: 0

N. J. A. Sloane, Mar 17 2004

nonn

A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004, p. 203.

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-2,0,3,-5,7,-7,4,-2,2,-1,-1,4,-6,6,-6,5,-3,-1,6,-6, 2,1,-4,5,-3,4,-5,2,-1,0,4,-6,9,-12,8,-2,-1,4,-6,6,-6,6,-6, 4,-1,-2,8,-12,9,-6,4,0,-1,2,-5,4,-3,5,-4,1,2,-6,6,-1,-3,5, -6,6,-6,4,-1,-1,2,-2,4,-7,7,-5,3,0,-2,2,-3,3,-1).

Cf. A082146, A089599, A091434, A091726.

CoefficientList[Series[(1-x+x^2-x^3+x^4-x^5+x^6-x^7+x^8-x^9+x^10) * (1-2*x+x^2+ x^5-x^6+x^10-x^11+2*x^12-2*x^13+x^14-x^15+x^16+x^17-x^18+x^19-x^21+2*x^22 - 2*x^23+3*x^24-2*x^25+2*x^26-x^27+x^29-x^30 +x^31+x^32-x^33+x^34-2*x^35+2*x^36 - x^37+x^38-x^42+x^43+x^46-2*x^47+x^48) / ((1-x)^3*(1-x^3)*(1-x^4)*(1-x^6)*(1- x^8)*(1-x^9)*(1-x^12)*(1-x^10)*(1-x^15)*(1-x^18)), {x,0,60}], x] (* G. C. Greubel, Jan 31 2020 *)

G.f.: ( x^75 + x^70 + x^68 + x^67 + x^66 + 2*x^65 + 2*x^64 + 2*x^63 + 3*x^62 + 4*x^61 + 4*x^60 + 5*x^59 + 5*x^58 + 6*x^57 + 7*x^56 + 8*x^55 + 10*x^54 + 10*x^53 + 11*x^52 + 13*x^51 + 14*x^50 + 15*x^49 + 17*x^48 + 18*x^47 + 19*x^46 + 20*x^45 + 21*x^44 + 22*x^43 + 23*x^42 + 23*x^41 + 24*x^40 + 23*x^39 + 24*x^38 + 24*x^37 + 23*x^36 + 24*x^35 + 23*x^34 + 23*x^33 + 22*x^32 + 21*x^31 + 20*x^30 + 19*x^29 + 18*x^28 + 17*x^27 + 15*x^26 + 14*x^25 + 13*x^24 + 11*x^23 + 10*x^22 + 10*x^21 + 8*x^20 + 7*x^19 + 6*x^18 + 5*x^17 + 5*x^16 + 4*x^15 + 4*x^14 + 3*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)^2 *(1 - x^10)*(1 - x^15)*(1 - x^18)).

cp's OEIS Frontend

A089599 G.f.: (1+x^5+x^7+x^8+x^10+x^15)/((1-x^2)(1-x^3)(1-x^4)(1-x^6)^2(1-x^9)).

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

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

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A091769 Poincaré series [or Poincare series] (or Molien series) for a certain six-fold wreath product P_6.

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