A091434 -id:A091434 - cp's OEIS frontend

A082146 Expansion of g.f.: (1+x^5)/((1-x^2)(1-x^3)(1-x^4)*(1-x^6)).

1, 0, 1, 1, 2, 2, 4, 3, 6, 6, 8, 9, 13, 12, 17, 18, 22, 24, 30, 30, 38, 40, 46, 50, 59, 60, 71, 75, 84, 90, 102, 105, 120, 126, 138, 147, 163, 168, 187, 196, 212, 224, 244, 252, 276, 288, 308, 324, 349, 360, 389, 405, 430, 450, 480, 495, 530, 550, 580, 605, 641, 660, 701, 726

Offset: 0

N. J. A. Sloane, Dec 30 2003

nonn

Poincaré series [or Poincare series] (or Molien series) for (P[x_0,x_1] ⊗ P[x_0,x_1])^(S_2).

A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 199.

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

Cf. A089599, A091434, A091726, A091769.

Cf. A010875 (n mod 6). Contains A006002 and A212683. - Luce ETIENNE, Aug 14 2018

R:=PowerSeriesRing(Integers(), 70);
Coefficients(R!( (1-x^10)/(&*[1-x^j: j in [2..6]]) )); // G. C. Greubel, Apr 02 2023

seq(coeff(series((1+x^5)/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^6)), x,n+1),x,n),n=0..70); # Muniru A Asiru, Aug 15 2018

CoefficientList[Series[(1-x^10)/Product[1-x^(j+1), {j,5}], {x,0,70}], x] (* G. C. Greubel, Apr 02 2023 *)

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

def A082146_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x^10)/prod(1-x^j for j in range(2,7)) ).list()
A082146_list(70) # G. C. Greubel, Apr 02 2023

a(n) = a(n-1) + a(n-3) - a(n-5) + a(n-6) - 2*a(n-7) + a(n-8) - a(n-9) + a(n-11) + a(n-13) - a(n-14).
G.f.: ( 1+x^2+x^4-x-x^3 ) / ( (1+x^2)*(1-x+x^2)*(1+x)^2*(1+x+x^2)^2*(1-x)^4 ). - R. J. Mathar, Oct 11 2011
a(n) = (120*floor(n/6)^3 + 60*(m+5)*floor(n/6)^2 - 20*(m^5-13*m^4 +60*m^3-116*m^2+74*m-18)*floor(n/6) - (19*m^5-245*m^4+1125*m^3-2185*m^2+1496*m-210) + (m^5-15*m^4+75*m^3-135*m^2+44*m+30)*(-1)^floor(n/6))/240 where m = (n mod 6). - Luce ETIENNE, Aug 14 2018

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)).

Original entry on oeis.org

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

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

A082146 Expansion of g.f.: (1+x^5)/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^6)).

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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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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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A082146 Expansion of g.f.: (1+x^5)/((1-x^2)(1-x^3)(1-x^4)*(1-x^6)).