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%I #22 Jan 30 2018 18:57:34
%S 1,0,0,1,1,1,2,2,2,3,3,4,5,5,6,7,7,8,10,10,11,13,13,14,16,17,18,20,21,
%T 22,24,25,27,29,30,32,34,35,37,40,41,43,46,47,49,52,54,56,59,61,63,66,
%U 68,71,74,76,79,82,84,87,91,93,96,100,102,105,109,112,115,119,122,125
%N G.f.: (1+x^3)*(1+x^5)*(1+x^6)/((1-x^4)*(1-x^6)*(1-x^7)) (or (1+x^5)(1+x^6)/((1-x^3)*(1-x^4)*(1-x^7))).
%C Poincaré series [or Poincare series] (or Molien series) for mod 2 cohomology of O'Nan group.
%C Also Poincaré series [or Poincare series] (or Molien series) for mod 2 cohomology of Janko group J_1.
%C Molien series of 3-dimensional representation of group of order 21 over GF(2).
%D A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004, pp. 77, 95 and 248.
%D D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 106.
%H A. Adem, <a href="http://www.ams.org/notices/199707/adem.pdf">Recent developments in the cohomology of finite groups</a>, Notices Amer. Math. Soc., 44 (1997), 806-812.
%H A. Adem and R. J. Milgram, <a href="http://dx.doi.org/10.1006/jabr.1995.1245">The subgroup structure and mod 2 cohomology of O'Nan's sporadic simple group</a>, J. Algebra 176 (1995), 288-315.
%H <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1,0,0,1,-1,0,-1,1).
%F G.f.: (x^4-x^2+1)*(x^4-x^3+x^2-x+1)/((1-x)*(1-x^3)*(1-x^7)). a(n)=a(n-3)+a(n-7)-a(n-10)+1, n>7.
%F G.f. can be written as q(x)/((1-x^8)(1-x^12)(1-x^14)) where q is a symmetric polynomial of degree 31 with nonnegative coefficients.
%p (1+x^3)*(1+x^5)*(1+x^6)/(1-x^4)/(1-x^6)/(1-x^7);
%t LinearRecurrence[{1, 0, 1, -1, 0, 0, 1, -1, 0, -1, 1}, {1, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3}, 80] (* _Vincenzo Librandi_, Jul 19 2015 *)
%o (PARI) Vec((1+x^3)*(1+x^5)*(1+x^6)/(1-x^4)/(1-x^6)/(1-x^7) + O(x^80)) \\ _Michel Marcus_, Jul 18 2015
%K nonn,easy
%O 0,7
%A _N. J. A. Sloane_
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