OFFSET
0,5
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).
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
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 *)
PROG
(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
(Magma)
R<x>:=PowerSeriesRing(Integers(), 70);
Coefficients(R!( (1-x^10)/(&*[1-x^j: j in [2..6]]) )); // G. C. Greubel, Apr 02 2023
(SageMath)
def A082146_list(prec):
P.<x> = 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
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Dec 30 2003
STATUS
approved