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%I #31 Feb 10 2022 06:56:00
%S 1,784,378,144452,92512,20475,11327232,8128792,2843568,376740,
%T 40116600,30421755,13123110,3108105,376740,20475,378,1,490068257,
%U 373673216,161446572,35904960,3108105,2096760960,1649657520,794670240,226546320,35904960,2843568,92512,784
%N 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.
%C 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
%C 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
%C Rows of length zero have simply been omitted.
%H John F. Duncan, <a href="https://arxiv.org/abs/math/0609449">Moonshine for Rudvalis's sporadic group I</a>, arXiv:math/0609449 [math.RT], November, 2008 (see pages 51-53)
%H John F. Duncan, <a href="https://arxiv.org/abs/math/0611355">Moonshine for Rudvalis's sporadic group II</a>, arXiv:math/0611355 [math.RT], November, 2008
%F 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
%e 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."
%e -------------------------------------------------------------------------|
%e .....|m=0...........|m=2..........|m=4..........|m=6.........|m=8........|
%e -------------------------------------------------------------------------|
%e n=..0|............1.|.............|.............|............|...........|
%e n=1/2|..............|.............|.............|............|...........|
%e n=..1|..........784.|.........378.|.............|............|...........|
%e n=3/2|..............|.............|.............|............|...........|
%e n=..2|.......144452.|.......92512.|.......20475.|............|...........|
%e n=.5/2|.............|.............|.............|............|...........|
%e n=...3|....11327232.|.....8128792.|.....2843568.|.....376740.|...........|
%e n=.7/2|....40116600.|....30421755.|....13123110.|....3108105.|.....376740|
%e n=...4|...490068257.|...373673216.|...161446572.|...35904960.|....3108105|
%e n=.9/2|..2096760960.|..1649657520.|...794670240.|..226546320.|...35904960|
%e n=...5|.13668945136.|.10818453324.|..5284484352.|.1513872360.|..226546320|
%e n=11/2|.56547022140.|.45624923820.|.23757475560.|.7766243940.|.1513872360|
%e -------------------------------------------------------------------------|
%Y Cf. A003918.
%K nonn,tabf
%O 0,2
%A _Jonathan Vos Post_, Nov 04 2008
%E Corrected by _Andrey Zabolotskiy_, Feb 18 2019
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