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%I #23 Jan 26 2020 20:58:20
%S 1,1,1,2,1,1,2,2,1,1,4,3,2,1,1,5,4,3,2,1,1,8,6,5,3,2,1,1,10,9,6,5,3,2,
%T 1,1,15,12,10,7,5,3,2,1,1,20,17,13,10,7,5,3,2,1,1,28,23,19,14,11,7,5,
%U 3,2,1,1,36,31,25,20,14,11,7,5,3,2,1,1,50,42
%N Triangle read by rows: T(n,k) (n >= 0, -1 <= k <= n-1) = number of partitions of n into nonnegative integer parts with rank k.
%C In contrast to A063995 and A105806, here we allow parts that are zeros.
%H Lars Blomberg, <a href="/A331447/b331447.txt">Table of n, a(n) for n = -1..5048</a>
%H Alexander Berkovich and Frank G. Garvan, <a href="https://doi.org/10.1006/jcta.2002.3281">Some observations on Dyson's new symmetries of partitions</a>, Journal of Combinatorial Theory, Series A 100.1 (2002): 61-93.
%H Freeman J. Dyson, <a href="https://doi.org/10.1016/S0021-9800(69)80006-2">A new symmetry of partitions</a>, Journal of Combinatorial Theory 7.1 (1969): 56-61. See Table 2.
%H Freeman J. Dyson, <a href="https://doi.org/10.1016/0097-3165(89)90043-5">Mappings and symmetries of partitions</a>, J. Combin. Theory Ser. A 51 (1989), 169-180.
%F See Dyson (1969).
%e Triangle begins:
%e 1,
%e 1, 1,
%e 2, 1, 1,
%e 2, 2, 1, 1,
%e 4, 3, 2, 1, 1,
%e 5, 4, 3, 2, 1, 1,
%e 8, 6, 5, 3, 2, 1, 1,
%e 10, 9, 6, 5, 3, 2, 1, 1,
%e ...
%e If we include negative values of the rank k, we get the following table, taken from Dyson (1969):
%e n\k| -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
%e ---+---------------------------------------------------
%e 0 | 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, ...
%e 1 | 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, ...
%e 2 | 2, 2, 2, 2, 2, 2, 1, 1, 0, 0, 0, 0, 0, ...
%e 3 | 3, 3, 3, 3, 3, 2, 2, 1, 1, 0, 0, 0, 0, ...
%e 4 | 5, 5, 5, 5, 4, 4, 3, 2, 1, 1, 0, 0, 0, ...
%e 5 | 7, 7, 7, 6, 6, 5, 4, 3, 2, 1, 1, 0, 0, ...
%e 6 | 11, 11, 10, 10, 9, 8, 6, 5, 3, 2, 1, 1, 0, ...
%e 7 | 15, 14, 14, 13, 12, 10, 9, 6, 5, 3, 2, 1, 1, ...
%e ...
%e Starting at column k=-1 gives the present triangle.
%Y For the rank of a partition see A063995, A105806.
%K nonn,tabl
%O -1,4
%A _N. J. A. Sloane_, Jan 23 2020
%E a(35) and beyond from _Lars Blomberg_, Jan 26 2020
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