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

Author

N. J. A. Sloane, Dec 30 2003

Keywords

• nonn

Comments

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

References

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

Links

• 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).

Crossrefs

Cf. A089599, A091434, A091726, A091769.

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

Programs

Magma

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

Maple

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

Mathematica

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

PARI

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

SageMath

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

Formula

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