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A008796 Molien series for 3-dimensional group [2,3]+ = 223; also for group H_{1,2} of order 384. 2

%I

%S 1,0,2,1,4,2,7,4,10,7,14,10,19,14,24,19,30,24,37,30,44,37,52,44,61,52,

%T 70,61,80,70,91,80,102,91,114,102,127,114,140,127,154,140,169,154,184,

%U 169,200,184,217,200,234,217,252,234,271,252,290,271,310,290,331,310,352,331,374,352,397

%N Molien series for 3-dimensional group [2,3]+ = 223; also for group H_{1,2} of order 384.

%H Vincenzo Librandi, <a href="/A008796/b008796.txt">Table of n, a(n) for n = 0..1000</a>

%H E. Bannai, S. T. Dougherty, M. Harada and M. Oura, <a href="https://sites.google.com/site/professorstevendougherty/publications">Type II Codes, Even Unimodular Lattices and Invariant Rings</a>, IEEE Trans. Information Theory, Volume 45, Number 4, 1999, 1194-1205.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,1,-1,-2,0,1).

%H <a href="/index/Mo#Molien">Index entries for Molien series</a>

%F G.f.: (1+x^4)/((1-x^2)^2*(1-x^3)).

%F a(n) = (1/72) * (9*(-1)^n*(2*n + 3) + 6*n^2 + 18*n + 29 - 8*A061347[n]). - _Ralf Stephan_, Apr 28 2014

%p seq(coeff(series((1+x^4)/((1-x^2)^2*(1-x^3)), x, n+1), x, n), n = 0..70); # _G. C. Greubel_, Sep 11 2019

%t LinearRecurrence[{0,2,1,-1,-2,0,1},{1,0,2,1,4,2,7},70] (* _Harvey P. Dale_, Apr 27 2014 *)

%t CoefficientList[Series[(1+x^4)/((1-x^2)^2*(1-x^3)), {x, 0, 70}], x] (* _Vincenzo Librandi_, Apr 28 2014 *)

%o (PARI) a(n)=(9*(-1)^n*(2*n + 3) + 6*n^2 + 18*n + 24*!(n%3) + 21)/72 \\ _Charles R Greathouse IV_, Feb 10 2017

%o (MAGMA) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (1+x^4)/((1-x^2)^2*(1-x^3)) )); // _G. C. Greubel_, Sep 11 2019

%o (Sage)

%o def A008796_list(prec):

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

%o return P((1+x^4)/((1-x^2)^2*(1-x^3))).list()

%o A008796_list(70) # _G. C. Greubel_, Sep 11 2019

%o (GAP) a:=[1,0,2,1,4,2,7];; for n in [8..70] do a[n]:=2*a[n-2]+a[n-3]-a[n-4]-2*a[n-5]+a[n-7]; od; a; # _G. C. Greubel_, Sep 11 2019

%Y Cf. A008795.

%K nonn,nice,easy

%O 0,3

%A _N. J. A. Sloane_

%E Definition clarified by _N. J. A. Sloane_, Feb 02 2018

%E More terms added by _G. C. Greubel_, Sep 11 2019

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Last modified December 7 20:33 EST 2019. Contains 329849 sequences. (Running on oeis4.)