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%I #25 Jul 15 2021 19:02:22
%S 1,1,3,7,21,47,128,303,754,1735,3989,8712,18687,38482,77421,150813,
%T 286925,531306,962637,1704506,2959412,5037606,8426351,13854300,
%U 22426944,35759968,56234440,87258555,133730542,202529129,303328391,449478982,659401717,958118335,1379571974,1969206260
%N Molien series for 16-dimensional group of structure 2^4.O_{4}^{+}(F_2) = 2^4.(S_3 X S_3).2 and order 1152, corresponding to genus 2 complete weight enumerators of Hermitian self-dual GF(2)-linear codes over GF(4) containing the all-ones vector.
%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. See Section (7.6.1), and especially Eq. (7.6.14) on page 219.
%H <a href="/index/Mo#Molien">Index entries for Molien series</a>
%F For the Molien series see the Maple code.
%p f1:= 1 + t^3 + 5*t^4 + 18*t^5 + 45*t^6 + 88*t^7 + 196*t^8 + 394*t^9 + 804*t^10 + 1512*t^11 + 2702*t^12 + 4529*t^13 + 7218*t^14 + 11019*t^15 + 16064*t^16 + 22609*t ^17 + 30555*t^18 + 39889*t^19 + 50303*t^20 + 61476*t^21 + 72888*t^22 + 84047*t^23 + 94299*t^24 + 102995*t^25 + 109674*t^26 + 113791*t^27 + 57614*t^28;
%p f:= f1+expand(t^56*subs(t=1/t,f1));
%p g:= (1-t)*(1-t^2)^2*(1-t^3)^3*(1-t^4)^6*(1-t^6)*(1-t^8)^2*(1-t^12);
%p h:=f/g; # This is the Molien series
%p series(h,t,48);
%Y Cf. A092201, A092497.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Apr 02 2004
%E There were errors in the definition (in the order and structure of the group). The rational form of the Molien series was correct, but the DATA section - the coefficients in the expansion of the Molien series - was wrong from the 28th term onwards. To make it easier to check I have replaced the formulas with Maple code based on the Latex source code for the book. Thanks to _Georg Fischer_ and _Andrey Zabolotskiy_ for noticing that something was wrong and proposing corrections. - _N. J. A. Sloane_, Jan 29 2021
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