A105319 Molien series for a certain 16-dimensional group of order 10321920.

1, 2, 4, 8, 20, 42, 95, 201, 465, 1014, 2292, 5014, 11043, 23594, 49910, 102699, 207520, 408597, 788081, 1484174, 2738264, 4944037, 8755686, 15208359, 25952790, 43527033, 71841425, 116752322, 187018934, 295451249, 460722893, 709561513, 1080061323

Offset

0,2

Links

Table of n, a(n) for n=0..32.

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.

Maple

u2:= (1-t^2)*(1-t^4)^2*(1-t^6)^2*(1-t^8)^4*(1-t^10)^2* (1-t^12)^2*(1-t^14)*(1-t^28)*(1-t^30); u2:=subs(t=sqrt(t), u2);
u1:= 1 + t + 3*t^4 + 9*t^5 + 24*t^6 + 46*t^ 7 + 117*t^8 + 239*t^9 + 541*t^10 + 1133 *t^11 + 2370*t^12 + 4649*t^13 + 8923*t^14 + 16245*t^15 + 28601*t^16 + 48132 *t^17 + 78194 *t^18 +121981 *t^19 +183920 *t^20 +267517 *t^21 + 376916 *t^22 + 514682 *t^23 + 683056*t^24 + 881972*t^25 + 1110910*t^26 +1366468 *t^27 + 1644918*t^28 + 1940048*t^29 + 2245177*t^30 + 2551867*t^31 + 2851403*t^32 + 3133830 *t^33 + 3389363*t^34 + 3608201 *t^35 + 3781448*t^36 + 3901399*t^37 + 3962896*t^38;
u1a:=expand(t^77*subs(t=1/t, u1)); u3:=(u1+u1a)/u2;

Prog

(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>;
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;
MM:=MatrixAlgebra(K, 16); hh:=TensorProduct(M ! 1, h); qq1:=TensorProduct(M ! 1, q1); pp1:=TensorProduct(M ! 1, p1); qq2:=TensorProduct(M ! 1, q2);
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])]; ML:=permstomats(L); UU:=MatrixGroup<16, K | hh, qq2, ML[1], ML[2], ML[3], ML[4]>; Order(UU); MUU:=MolienSeries(UU);

Crossrefs

Sequence in context: A236397 A272122 A323019 * A051389 A078006 A288476

Adjacent sequences: A105316 A105317 A105318 * A105320 A105321 A105322

Keyword

nonn

Author

N. J. A. Sloane and Gabriele Nebe, Apr 26 2005

Status

approved