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A092091 Molien series for 9-dimensional group of structure Z_2 X Z_2 and order 4, corresponding to complete weight enumerators of Hermitian self-dual GF(3)-linear codes over GF(9).

%I A092091 #24 Sep 08 2022 08:45:13
%S A092091 1,4,17,52,147,360,819,1712,3382,6312,11286,19368,32154,51744,81114,
%T A092091 124080,185823,272844,393679,558844,781781,1078792,1470261,1980576,
%U A092091 2639676,3482960,4553212,5900496,7584516,9674496,12252036,15410976,19260813,23926548,29552733
%N A092091 Molien series for 9-dimensional group of structure Z_2 X Z_2 and order 4, corresponding to complete weight enumerators of Hermitian self-dual GF(3)-linear codes over GF(9).
%H A092091 Colin Barker, <a href="/A092091/b092091.txt">Table of n, a(n) for n = 0..1000</a>
%H A092091 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 A092091 <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H A092091 <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (5,-6,-10,29,-9,-36,36,9,-29,10,6,-5,1).
%F A092091 G.f.: (1 +2*x^2 +4*x^3 +x^4)/((1-x)^4*(1-x^2)^5).
%F A092091 G.f.: (1 -x +3*x^2 +x^3)/( (1+x)^4*(1-x)^9 ). - _R. J. Mathar_, Dec 18 2014
%F A092091 a(n) = ((315*(857+167*(-1)^n) + 60*(8347+581*(-1)^n)*n + (384718+6930*(-1)^n)*n^2 + 84*(2027+5*(-1)^n)*n^3 + 48888*n^4 + 9240*n^5 + 1092*n^6 + 72*n^7 + 2*n^8)) / 322560. - _Colin Barker_, Jan 16 2017
%p A092091 seq(coeff(series((1-x+3*x^2+x^3)/((1+x)^4*(1-x)^9), x, n+1), x, n), n = 0..40); # _G. C. Greubel_, Feb 02 2020
%t A092091 LinearRecurrence[{5,-6,-10,29,-9,-36,36,9,-29,10,6,-5,1}, {1,4,17,52,147,360, 819,1712,3382,6312,11286,19368,32154}, 35] (* _Ray Chandler_, Jul 15 2015 *)
%o A092091 (PARI) Vec((1 -x +3*x^2 +x^3)/((1-x)^9*(1+x)^4) + O(x^40)) \\ _Colin Barker_, Jan 16 2017
%o A092091 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x+3*x^2+x^3)/((1+x)^4*(1-x)^9) )); // _G. C. Greubel_, Feb 02 2020
%o A092091 (Sage)
%o A092091 def A092091_list(prec):
%o A092091     P.<x> = PowerSeriesRing(ZZ, prec)
%o A092091     return P( (1-x+3*x^2+x^3)/((1+x)^4*(1-x)^9) ).list()
%o A092091 A092091_list(40) # _G. C. Greubel_, Feb 02 2020
%o A092091 (GAP) List([0..40], n-> ((315*(857 +167*(-1)^n) +60*(8347 +581*(-1)^n)*n + (384718 +6930*(-1)^n)*n^2 +84*(2027 +5*(-1)^n)*n^3 +48888*n^4 +9240*n^5 +1092*n^6 +72*n^7 +2*n^8))/322560 ); # _G. C. Greubel_, Feb 02 2020
%Y A092091 Cf. A052365.
%K A092091 nonn,easy
%O A092091 0,2
%A A092091 _N. J. A. Sloane_, Apr 01 2004