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%I #37 Sep 02 2023 11:02:40
%S 1,1,1,1,2,2,3,3,4,5,6,6,9,10,11,12,15,16,19,20,23,26,29,30,36,39,42,
%T 45,51,54,60,63,69,75,81,84,94,100,106,112,122,128,138,144,154,164,
%U 174,180,195,205,215,225,240,250,265,275,290,305,320,330,351,366
%N Expansion of g.f.: (1+x^9)/((1-x)*(1-x^4)*(1-x^6)*(1-x^12)).
%C Molien series for genus-2 weight enumerators of binary self-dual codes is (1+x^18)/((1-x^2)*(1-x^8)*(1-x^12)*(1-x^24)). Exponents have been divided by 2 to get the sequence.
%C Or, Molien series for 4-dimensional representation of 2.{3,4,3}. This is the real 4-dimensional Clifford group of genus 2 and order 2304.
%H T. D. Noe, <a href="/A008718/b008718.txt">Table of n, a(n) for n = 0..1000</a>
%H F. J. MacWilliams, C. L. Mallows and N. J. A. Sloane, <a href="http://neilsloane.com/doc/gleason2.html">Generalizations of Gleason's theorem on weight enumerators of self-dual codes</a>, IEEE Trans. Inform. Theory, 18 (1972), 794-805; see p. 802, col. 2, foot.
%H G. Nebe, E. M. Rains and N. J. A. Sloane, <a href="http://neilsloane.com/doc/cliff2.html">Self-Dual Codes and Invariant Theory</a>, Springer, Berlin, 2006.
%H <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H <a href="/index/Rec#order_20">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,0,-1,0,-1,1,0,0,0,1,-1,0,-1,0,1,0,1,-1).
%F a(n) ~ (1/864)*n^3. - _Ralf Stephan_, Apr 29 2014
%F G.f.: ( 1-x^3+x^6 ) / ( (1-x+x^2) *(x^4-x^2+1) *(1+x)^2 *(x^2+1)^2 *(1+x+x^2)^2 *(x-1)^4 ). - _R. J. Mathar_, Dec 18 2014
%p (1+x^9)/((1-x)*(1-x^4)*(1-x^6)*(1-x^12)); seq(coeff(series(%, x, n+1), x, n), n = 0..65); # modified by _G. C. Greubel_, Sep 09 2019
%t CoefficientList[Series[(1+x^9)/((1-x)(1-x^4)(1-x^6)(1-x^12)), {x,0,65}], x] (* _Harvey P. Dale_, Apr 01 2011 *)
%t LinearRecurrence[{1,0,1,0,-1,0,-1,1,0,0,0,1,-1,0,-1,0,1,0,1,-1}, {1,1,1, 1,2,2,3,3,4,5,6,6,9,10,11,12,15,16,19,20}, 65] (* _Ray Chandler_, Jul 16 2015 *)
%o (PARI) my(x='x+O('x^65)); Vec((1+x^9)/((1-x)*(1-x^4)*(1-x^6)*(1-x^12))) \\ _G. C. Greubel_, Sep 09 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 65); Coefficients(R!( (1+x^9)/((1-x)*(1-x^4)*(1-x^6)*(1-x^12)) )); // _G. C. Greubel_, Sep 09 2019
%o (Sage)
%o def A008718_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P((1+x^9)/((1-x)*(1-x^4)*(1-x^6)*(1-x^12))).list()
%o A008718_list(65) # _G. C. Greubel_, Sep 09 2019
%Y Cf. A008621, A024186.
%K nonn,easy,nice
%O 0,5
%A _N. J. A. Sloane_
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