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%I #20 Sep 08 2022 08:45:13
%S 1,1,2,3,6,9,17,27,49,79,138,225,388,634,1069,1750,2904,4706,7677,
%T 12302,19715,31137,48994,76133,117637,179718,272692,409558,610615,
%U 901812,1322019,1921168,2771490,3965936,5635172,7947036,11132032,15484913,21402697,29390457
%N Molien series for certain 16-dimensional group of order 322560 arising from genus-2 weight enumerators of singly-even (but not necessarily self-dual or self-orthogonal) binary codes.
%H Robert Israel, <a href="/A092351/b092351.txt">Table of n, a(n) for n = 0..10000</a>
%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/Rec#order_96">Index entries for linear recurrences with constant coefficients</a>, signature (2, 1, -2, -2, 0, 2, -1, 3, -1, 0, 0, 3, -4, -5, 4, -4, 1, 2, 11, -5, 1, -2, -5, -1, -2, 0, -8, 19, 2, 2, -6, 3, -10, -7, 6, -11, 10, 6, 13, -10, 12, -6, -19, -4, 3, 7, -7, 20, -7, 7, 3, -4, -19, -6, 12, -10, 13, 6, 10, -11, 6, -7, -10, 3, -6, 2, 2, 19, -8, 0, -2, -1, -5, -2, 1, -5, 11, 2, 1, -4, 4, -5, -4, 3, 0, 0, -1, 3, -1, 2, 0, -2, -2, 1, 2, -1).
%F G.f. = f(x)/g(x) where
%F f(x) = x^80 - x^79 - x^78 + 2*x^76 + x^74 + x^73 + 2*x^72 + 5*x^70 + 3*x^69 + 13*x^68 + 10*x^67 + 29*x^66 + 31*x^65 + 66*x^64 + 61*x^63 + 121*x^62 + 138*x^61 + 232*x^60 + 258*x^59 + 396*x^58 + 456*x^57 + 654*x^56 + 739*x^55 + 974*x^54 + 1098*x^53 + 1400*x^52 + 1529*x^51 + 1859*x^50 + 2013*x^49 + 2363*x^48 + 2486*x^47 + 2831*x^46 + 2923*x^45 + 3222*x^44 + 3233*x^43 + 3461*x^42 + 3406*x^41 + 3570*x^40 + 3406*x^39 + 3461*x^38 + 3233*x^37 + 3222*x^36 + 2923*x^35 + 2831*x^34 + 2486*x^33 + 2363*x^32 + 2013*x^31 + 1859*x^30 + 1529*x^29 + 1400*x^28 + 1098*x^27 + 974*x^26 + 739*x^25 + 654*x^24 + 456*x^23 + 396*x^22 + 258*x^21 + 232*x^20 + 138*x^19 + 121*x^18 + 61*x^17 + 66*x^16 + 31*x^15 + 29*x^14 + 10*x^13 + 13*x^12 + 3*x^11 + 5*x^10 + 2*x^8 + x^7 + x^6 + 2*x^4 - x^2 - x + 1 and
%F g(x) = x^96 - 2*x^95 - x^94 + 2*x^93 + 2*x^92 - 2*x^90 + x^89 - 3*x^88 + x^87 - 3*x^84 + 4*x^83 + 5*x^82 - 4*x^81 + 4*x^80 - x^79 - 2*x^78 - 11*x^77 + 5*x^76 - x^75 + 2*x^74 + 5*x^73 + x^72 + 2*x^71 + 8*x^69 - 19*x^68 - 2*x^67 - 2*x^66 + 6*x^65 - 3*x^64 + 10*x^63 + 7*x^62 - 6*x^61 + 11*x^60 - 10*x^59 - 6*x^58 - 13*x^57 + 10*x^56 - 12*x^55 + 6*x^54 + 19*x^53 + 4*x^52 - 3*x^51 - 7*x^50 + 7*x^49 - 20*x^48 + 7*x^47 - 7*x^46 - 3*x^45 + 4*x^44 + 19*x^43 + 6*x^42 - 12*x^41 + 10*x^40 - 13*x^39 - 6*x^38 - 10*x^37 + 11*x^36 - 6*x^35 + 7*x^34 + 10*x^33 - 3*x^32 + 6*x^31 - 2*x^30 - 2*x^29 - 19*x^28 + 8*x^27 + 2*x^25 + x^24 + 5*x^23 + 2*x^22 - x^21 + 5*x^20 - 11*x^19 - 2*x^18 - x^17 + 4*x^16 - 4*x^15 + 5*x^14 + 4*x^13 - 3*x^12 + x^9 - 3*x^8 + x^7 - 2*x^6 + 2*x^4 + 2*x^3 - x^2 - 2*x + 1.
%o (Magma) K:=Rationals(); M:=MatrixAlgebra(K,4); q1:=DiagonalMatrix(M,[1,-1,1,-1]); p1:=DiagonalMatrix(M,[1,1,-1,-1]); q2:=DiagonalMatrix(M,[1,1,1,-1]); h:=M![1,1,1,1, 1,1,-1,-1, 1,-1,1,-1, 1,-1,-1,1]/2; H:=MatrixGroup<4,K|q1,q2,h,p1>;
%o permstomats:=function(L); n:=#L[1]; M:=MatrixAlgebra(Rationals(),n); a:=#L; MM:=[]; for i in [1..a] do Append(~MM,M ! 0); end for; for i in [1..a] do for j in [1..n] do MM[i][j][L[i][j]]:=1; end for; end for; return MM; end function;
%o MM:=MatrixAlgebra(K,16); hh:=TensorProduct(M ! 1,h); qq1:=TensorProduct(M ! 1,q1); pp1:=TensorProduct(M ! 1,p1); qq2:=TensorProduct(M ! 1,q2);
%o perm:=sub<Sym(16) | (3,5)*(4,6)*(11,13)*(12,14), (3,7)*(4,8)*(11,15)*(12,16), (2,10)*(4,12)*(6,14)*(8,16),(2,9)*(4,11)*(6,13)*(8,15)>; Order(perm); pp:=Setseq(Generators(perm)); L:=[Eltseq(pp[1]),Eltseq(pp[2]),Eltseq(pp[3]),Eltseq(pp[4])];
%o ML:=permstomats(L); GG:=MatrixGroup<16,K | hh,qq1,qq2,ML[1],ML[2],ML[3],ML[4]>; Order(GG); MGG:=MolienSeries(GG);
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Mar 20 2004
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