%I #30 Dec 01 2023 03:26:00
%S 1,1,1,1,2,2,3,4,6,7,10,12,18,22,29,35,48,57,74,91,116,140,177,211,
%T 265,319,389,462,566,667,804,949,1131,1324,1573,1827,2153,2502,2917,
%U 3364,3916,4491,5187,5937,6813,7760,8879,10058,11448,12950,14658,16500,18632,20894
%N Expansion of Molien series for 8-dimensional real Clifford group 2^{1+6}.Alt_8.2 of genus 3 and order 5160960.
%H Jean-François Alcover, <a href="/A024186/b024186.txt">Table of n, a(n) for n = 0..999</a>
%H A. R. Calderbank, R. H. Hardin, E. M. Rains, P. W. Shor, and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0208002">A Group-Theoretic Framework for the Construction of Packings in Grassmannian Spaces</a>, J. Algebraic Combinatorics, 9 (1999), 129-140; arXiv:math/0208002 [math.CO], 2002.
%H A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, <a href="https://arxiv.org/abs/quant-ph/9608006">Quantum Error Correction Via Codes Over GF(4)</a>, arXiv:quant-ph/9608006, 1996-1997; IEEE Trans. Information Theory, 44 (1998), 1369-1387.
%H J. H. Conway, R. H. Hardin, and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0208004">Packing Lines, Planes, etc.: Packings in Grassmannian Space</a>, Experimental Math. 5 (1996), 139-159; arXiv:math/0208004 [math.CO], 2002.
%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/Mo#Molien">Index entries for Molien series</a>
%H <a href="/index/Rec#order_78">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 1, -1, 0, 0, 1, 0, -1, -1, 0, 3, -2, 0, -2, 2, 0, 0, -2, 1, 1, 1, -1, -1, 2, 0, -1, -1, 1, 2, 0, -3, 0, 2, 2, -3, -2, 0, 4, 0, -2, -3, 2, 2, 0, -3, 0, 2, 1, -1, -1, 0, 2, -1, -1, 1, 1, 1, -2, 0, 0, 2, -2, 0, -2, 3, 0, -1, -1, 0, 1, 0, 0, -1, 1, 0, 1, -1).
%F Molien series = (t^148 - t^142 + t^140 + t^136 - t^134 + t^132 + 3*t^128 + 2*t^124 + 4*t^120 + 5*t^116 + 7*t^112 + t^110 + 7*t^108 + t^106 + 10*t^104 + 2*t^102 + 11*t^100 + 3*t^98 + 12*t^96 + 4*t^94 + 14*t^92 + 5*t^90 + 16*t^88 + 5*t^86 + 15*t^84 + 4*t^82 + 20*t^80 + 7*t^78 + 18*t^76 + 4*t^74 + 18*t^72 + 7*t^70 + 20*t^68 + 4*t^66 + 15*t^64 + 5*t^62 + 16*t^60 + 5*t^58 + 14*t^56 + 4*t^54 + 12*t^52 + 3*t^50 + 11*t^48 + 2*t^46 + 10*t^44 + t^42 + 7*t^40 + t^38 + 7*t^36 + 5*t^32 + 4*t^28 + 2*t^24 + 3*t^20 + t^16 - t^14 + t^12 + t^8 - t^6 + 1) /
%F (t^156 - t^154 - t^150 + t^148 - t^142 + t^138 + t^136 - 3*t^132 + 2*t^130 + 2*t^126 - 2*t^124 + 2*t^118 - t^116 - t^114 - t^112 + t^110 + t^108 - 2*t^106 + t^102 + t^100 - t^98 - 2*t^96 + 3*t^92 - 2*t^88 - 2*t^86 + 3*t^84 + 2*t^82 - 4*t^78 + 2*t^74 + 3*t^72 - 2*t^70 - 2*t^68 + 3*t^64 - 2*t^60 - t^58 + t^56 + t^54 - 2*t^50 + t^48 + t^46 - t^44 - t^42 - t^40 + 2*t^38 - 2*t^32 + 2*t^30 + 2*t^26 - 3*t^24 + t^20 + t^18 - t^14 + t^8 - t^6 - t^2 + 1).
%t ker = {1, 0, 1, -1, 0, 0, 1, 0, -1, -1, 0, 3, -2, 0, -2, 2, 0, 0, -2, 1, 1, 1, -1, -1, 2, 0, -1, -1, 1, 2, 0, -3, 0, 2, 2, -3, -2, 0, 4, 0, -2, -3, 2, 2, 0, -3, 0, 2, 1, -1, -1, 0, 2, -1, -1, 1, 1, 1, -2, 0, 0, 2, -2, 0, -2, 3, 0, -1, -1, 0, 1, 0, 0, -1, 1, 0, 1, -1};
%t init = {1, 1, 1, 1, 2, 2, 3, 4, 6, 7, 10, 12, 18, 22, 29, 35, 48, 57, 74, 91, 116, 140, 177, 211, 265, 319, 389, 462, 566, 667, 804, 949, 1131, 1324, 1573, 1827, 2153, 2502, 2917, 3364, 3916, 4491, 5187, 5937, 6813, 7760, 8879, 10058, 11448, 12950, 14658, 16500, 18632, 20894, 23487, 26279, 29417, 32801, 36630, 40695, 45285, 50223, 55690, 61559, 68119, 75092, 82841, 91141, 100256, 110026, 120800, 132226, 144804, 158251, 172881, 188489, 205560, 223657};
%t LinearRecurrence[ker, init, 1000] (* _Jean-François Alcover_, Jan 05 2020 *)
%o (Magma)
%o // Commands to generate the group.
%o F<s> := QuadraticField(2); M := GeneralLinearGroup(8, F); t := 1/(2*s);
%o B := M! [ -t, -t, -t, -t, -t, -t, -t, -t,
%o -t, t, -t, -t, t, -t, t, t,
%o -t, -t, -t, t, -t, t, t, t,
%o -t, -t, t, -t, t, t, t, -t,
%o -t, t, -t, t, t, t, -t, -t,
%o -t, -t, t, t, t, -t, -t, t,
%o -t, t, t, t, -t, -t, t, -t,
%o -t, t, t, -t, -t, t, -t, t ];
%o S := M! [ -1, 0, 0, 0, 0, 0, 0, 0,
%o 0, -1, 0, 0, 0, 0, 0, 0,
%o 0, 0, 1, 0, 0, 0, 0, 0,
%o 0, 0, 0, 1, 0, 0, 0, 0,
%o 0, 0, 0, 0, 1, 0, 0, 0,
%o 0, 0, 0, 0, 0, 1, 0, 0,
%o 0, 0, 0, 0, 0, 0, 1, 0,
%o 0, 0, 0, 0, 0, 0, 0, 1 ];
%o C := M! [ 1, 0, 0, 0, 0, 0, 0, 0,
%o 0, 0, 1, 0, 0, 0, 0, 0,
%o 0, 0, 0, 1, 0, 0, 0, 0,
%o 0, 0, 0, 0, 1, 0, 0, 0,
%o 0, 0, 0, 0, 0, 1, 0, 0,
%o 0, 0, 0, 0, 0, 0, 1, 0,
%o 0, 0, 0, 0, 0, 0, 0, 1,
%o 0, 1, 0, 0, 0, 0, 0, 0 ];
%o G := sub< M | B, S, C >; Order(G);
%Y Cf. A008621, A008718.
%K nonn,easy,nice
%O 0,5
%A _N. J. A. Sloane_, G. Nebe (nebe(AT)math.rwth-aachen.de)
%E Rechecked Mar 30 2004. There were errors in the formula line, although not in the sequence itself.