A091972 Expansion of g.f. (1 + x^5 ) / ( (1-x^3)*(1-x^4)).

1, 0, 0, 1, 1, 1, 1, 1, 2, 2, 1, 2, 3, 2, 2, 3, 3, 3, 3, 3, 4, 4, 3, 4, 5, 4, 4, 5, 5, 5, 5, 5, 6, 6, 5, 6, 7, 6, 6, 7, 7, 7, 7, 7, 8, 8, 7, 8, 9, 8, 8, 9, 9, 9, 9, 9, 10, 10, 9, 10, 11, 10, 10, 11, 11, 11, 11, 11, 12, 12, 11, 12, 13, 12, 12, 13, 13, 13, 13, 13, 14, 14, 13, 14, 15, 14, 14, 15, 15, 15

Offset: 0

Author

N. J. A. Sloane, Mar 18 2004

Keywords

• nonn
• easy
• changed

Comments

Poincaré series [or Poincare series] (or Molien series) for Mathieu group M_11.

References

• A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 247.

Links

• Index entries for linear recurrences with constant coefficients, signature (1,-1,2,-1,1,-1).

Programs

Mathematica

CoefficientList[Series[(1+x^5)/((1-x^3)(1-x^4)),{x,0,90}],x] (* or *) LinearRecurrence[{1,-1,2,-1,1,-1},{1,0,0,1,1,1},90] (* Harvey P. Dale, Dec 11 2012 *)

Formula

a(0)=1, a(1)=0, a(2)=0, a(3)=1, a(4)=1, a(5)=1, a(n)=a(n-1)-a(n-2)+2*a(n-3)-a(n-4)+a(n-5)-a(n-6) for n > 5. - Harvey P. Dale, Dec 11 2012

G.f.: ( 1-x^3-x+x^2+x^4 ) / ( (x^2+1)*(1+x+x^2)*(x-1)^2 ). - R. J. Mathar, Sep 27 2014

E.g.f.: (3*exp(x)*(1 + x) + 9*cos(x) + exp(-x/2)*(6*cos(sqrt(3)*x/2) - 2*sqrt(3)*sin(sqrt(3)*x/2)))/18. - Stefano Spezia, Aug 26 2025