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.

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

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

Table of n, a(n) for n=0..30.

John F. Duncan, Moonshine for Rudvalis's sporadic group I, arXiv:math/0609449 [math.RT], November, 2008 (see pages 51-53)

John F. Duncan, Moonshine for Rudvalis's sporadic group II, arXiv:math/0611355 [math.RT], November, 2008

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

Cf. A003918.

Sequence in context: A204279 A158399 A007243 * A095954 A351476 A257355

Adjacent sequences: A146975 A146976 A146977 * A146979 A146980 A146981

Keyword

nonn,tabf

Author

Jonathan Vos Post, Nov 04 2008

Extensions

Corrected by Andrey Zabolotskiy, Feb 18 2019

Status

approved