|
| |
|
|
A146978
|
|
Irregular triangle read by rows: coefficients of the two-variable character of the vertex operator superalgebra A_Ru related to the sporadic simple Rudvalis group.
|
|
0
|
|
|
|
1, 784, 378, 144452, 92512, 20475, 11327232, 8128792, 2843568, 376740, 40116600, 30421755, 13123110, 3108105, 376740, 20475, 378, 1, 490068257, 373673216, 161446572, 35904960, 3108105, 2096760960, 1649657520, 794670240, 226546320, 35904960, 2843568, 92512, 784
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
Note that A_Ru is not the Rudvalis sporadic simple group Ru, rather it is a certain vertex operator superalgebra of rank 28 whose full automorphism group is a direct product of a cyclic group of order seven with Ru. - N. J. A. Sloane, Sep 17 2020
The row index n (the "degree") runs through nonnegative integers and half-integers, while the column index m (the "charge") runs through a finite number of nonnegative even integers (see the table in the Example section). If n is an integer row n has length n+1 (so the maximal index is m=2n); for half-integers n = 1/2, 3/2, 5/2, ... the row lengths are 0, 0, 0, 8, 8, 9, 9, 10, 10... (the pattern of repeated consecutive integers seems to continue). - Andrey Zabolotskiy, Sep 23 2020
Rows of length zero have simply been omitted.
|
|
|
LINKS
|
|
|
|
FORMULA
|
T(n+7/2, 2*(7-k)) = T(n+k, 2*k) = T(n+14-k, 28-2*k) for n = 0..3, k = 0..7. Also, T(k, 2*k) = binomial(28, 2*k). - Andrey Zabolotskiy, Feb 18 2019
|
|
|
EXAMPLE
|
Duncan: "The column headed m is the coefficient of p^m (as a series in q) and the row headed n is the coefficient of q^(n-c/24) (as a series in p). The coefficients of p^-m and p^m coincide and all subspaces of odd charge vanish."
-------------------------------------------------------------------------|
.....|m=0...........|m=2..........|m=4..........|m=6.........|m=8........|
-------------------------------------------------------------------------|
n=..0|............1.|.............|.............|............|...........|
n=1/2|..............|.............|.............|............|...........|
n=..1|..........784.|.........378.|.............|............|...........|
n=3/2|..............|.............|.............|............|...........|
n=..2|.......144452.|.......92512.|.......20475.|............|...........|
n=.5/2|.............|.............|.............|............|...........|
n=...3|....11327232.|.....8128792.|.....2843568.|.....376740.|...........|
n=.7/2|....40116600.|....30421755.|....13123110.|....3108105.|.....376740|
n=...4|...490068257.|...373673216.|...161446572.|...35904960.|....3108105|
n=.9/2|..2096760960.|..1649657520.|...794670240.|..226546320.|...35904960|
n=...5|.13668945136.|.10818453324.|..5284484352.|.1513872360.|..226546320|
n=11/2|.56547022140.|.45624923820.|.23757475560.|.7766243940.|.1513872360|
-------------------------------------------------------------------------|
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,tabf
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
