OFFSET
0,4
COMMENTS
Poincaré series [or Poincare series] (or Molien series) for mod 2 cohomology of M_12.
REFERENCES
A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 255, Theorem 3.20, where the series is given in the form GF_2 (see formula line).
LINKS
T. D. Noe, Table of n, a(n) for n = 0..1000
A. Adem, Recent developments in the cohomology of finite groups, Notices Amer. Math. Soc., 44 (1997), 806-812.
Alejandro Adem; John Maginnis; James R. Milgram, The geometry and cohomology of the Mathieu group M_12, J. Algebra 139 (1991), no. 1, 90-133.
Index entries for linear recurrences with constant coefficients, signature (2,-2,3,-3,2,-2,1).
FORMULA
Can also be written as GF_2 = (1 + x^2 + 3*x^3 + x^4 + 3*x^5 + 4*x^6 + 2*x^7 + 4*x^8 + 3*x^9 + x^10 + 3*x^11 + x^12 + x^14 ) / ( (1-x^4)*(1-x^6)*(1-x^7)).
G.f.: (1-x+x^2)^2/((1-x)^3*(1+x^2)(1+x+x^2)). a(n)=n^2/12+n/4+13/36-A057077(n)/4+4*A099837(n+3)/9. - R. J. Mathar, Jan 11 2009
a(0)=1, a(1)=0, a(2)=1, a(3)=3, a(4)=2, a(5)=3, a(6)=6, a(n)= 2*a(n-1)- 2*a(n-2)+3*a(n-3)-3*a(n-4)+2*a(n-5)-2*a(n-6)+a(n-7). - Harvey P. Dale, Apr 10 2012
MAPLE
(1+x^3)^2/((1-x^2)*(1-x^3)*(1-x^4));
MATHEMATICA
CoefficientList[Series[(1+x^3)^2/((1-x^2)*(1-x^3)*(1-x^4)), {x, 0, 60}], x] (* Harvey P. Dale, Mar 17 2011 *)
LinearRecurrence[{2, -2, 3, -3, 2, -2, 1}, {1, 0, 1, 3, 2, 3, 6}, 60] (* Harvey P. Dale, Apr 10 2012 *)
PROG
(Magma) I:=[1, 0, 1, 3, 2, 3, 6]; [n le 7 select I[n] else 2*Self(n-1)- 2*Self(n-2)+3*Self(n-3)-3*Self(n-4)+2*Self(n-5)-2*Self(n-6)+Self(n-7): n in [1..60]]; // Vincenzo Librandi, Jul 19 2015
(PARI) a(n)=([0, 1, 0, 0, 0, 0, 0; 0, 0, 1, 0, 0, 0, 0; 0, 0, 0, 1, 0, 0, 0; 0, 0, 0, 0, 1, 0, 0; 0, 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 0, 1; 1, -2, 2, -3, 3, -2, 2]^n*[1; 0; 1; 3; 2; 3; 6])[1, 1] \\ Charles R Greathouse IV, Feb 10 2017
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved