A024186 Expansion of Molien series for 8-dimensional real Clifford group 2^{1+6}.Alt_8.2 of genus 3 and order 5160960.
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
Offset: 0
Links
- Jean-François Alcover, Table of n, a(n) for n = 0..999
- A. R. Calderbank, R. H. Hardin, E. M. Rains, P. W. Shor, and N. J. A. Sloane, A Group-Theoretic Framework for the Construction of Packings in Grassmannian Spaces, J. Algebraic Combinatorics, 9 (1999), 129-140; arXiv:math/0208002 [math.CO], 2002.
- A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, Quantum Error Correction Via Codes Over GF(4), arXiv:quant-ph/9608006, 1996-1997; IEEE Trans. Information Theory, 44 (1998), 1369-1387.
- J. H. Conway, R. H. Hardin, and N. J. A. Sloane, Packing Lines, Planes, etc.: Packings in Grassmannian Space, Experimental Math. 5 (1996), 139-159; arXiv:math/0208004 [math.CO], 2002.
- G. Nebe, E. M. Rains, and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
- Index entries for Molien series
- Index entries for linear recurrences with constant coefficients, 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).
Programs
-
Magma
// Commands to generate the group. F
:= QuadraticField(2); M := GeneralLinearGroup(8, F); t := 1/(2*s); B := M! [ -t, -t, -t, -t, -t, -t, -t, -t, -t, t, -t, -t, t, -t, t, t, -t, -t, -t, t, -t, t, t, t, -t, -t, t, -t, t, t, t, -t, -t, t, -t, t, t, t, -t, -t, -t, -t, t, t, t, -t, -t, t, -t, t, t, t, -t, -t, t, -t, -t, t, t, -t, -t, t, -t, t ]; S := M! [ -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1 ]; C := M! [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0 ]; G := sub< M | B, S, C >; Order(G); -
Mathematica
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}; 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}; LinearRecurrence[ker, init, 1000] (* Jean-François Alcover, Jan 05 2020 *)
Formula
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) /
(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).
Extensions
Rechecked Mar 30 2004. There were errors in the formula line, although not in the sequence itself.