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A105319 Molien series for a certain 16-dimensional group of order 10321920. 0
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified April 14 04:41 EDT 2021. Contains 342941 sequences. (Running on oeis4.)