[seqfan] Re: Regarding A122172 - A proper definition is needed for this sequence

[franktaw at netscape.net]

This can be regarded as a tabf sequence in two different ways. One has 
one row per rectangular array, so nth row has (n+1)*Part(n) elements.

The other way is with one row of a table counting as a row of the 
sequence. Thus this produces row lengths of 1, 2, 3,3, 4,4,4, 
5,5,5,5,5, ....

Franklin T. Adams-Watters

-----Original Message-----
From: David Scambler <dscambler at bmm.com>

Part of the problem with presenting this sequence is that it is an 
infinite
sequence of rectangular arrays, not of integers.

The j'th array in the triangle has j+1 columns and partitions(j) rows.

Each row of the j'th section corresponds to a partition of j and 
answers the
following question:

How many ways may 0<=k<=j indistinguishable balls be dropped into a set 
of urns
whose maximum capacities
are determined by the parts of a partition of j?

e.g. j = 3

Urns   k= 0 1 2 3
========  =======
(3)       1 1 1 1
(2)(1)    1 2 2 1
(1)(1)(1) 1 3 3 1

dave

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