A082146 - OEIS

%I #32 Apr 03 2023 09:16:38

%S 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,

%T 75,84,90,102,105,120,126,138,147,163,168,187,196,212,224,244,252,276,

%U 288,308,324,349,360,389,405,430,450,480,495,530,550,580,605,641,660,701,726

%N Expansion of g.f.: (1+x^5)/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^6)).

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

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

%H Ray Chandler, <a href="/A082146/b082146.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,0,-1,1,-2,1,-1,0,1,0,1,-1).

%F 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).

%F 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

%F 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

%p 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

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

%o (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

%o (Magma)

%o R<x>:=PowerSeriesRing(Integers(), 70);

%o Coefficients(R!( (1-x^10)/(&*[1-x^j: j in [2..6]]) )); // _G. C. Greubel_, Apr 02 2023

%o (SageMath)

%o def A082146_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( (1-x^10)/prod(1-x^j for j in range(2,7)) ).list()

%o A082146_list(70) # _G. C. Greubel_, Apr 02 2023

%Y Cf. A089599, A091434, A091726, A091769.

%Y Cf. A010875 (n mod 6). Contains A006002 and A212683. - _Luce ETIENNE_, Aug 14 2018

%K nonn

%O 0,5

%A _N. J. A. Sloane_, Dec 30 2003