[seqfan] balls of u distinct colors filling u unlabeled urns of capacity b

[Richard J. Mathar]

This here is a combinatorial enumeration problem, where I'm wondering whether 
this is in the OEIS:
Let there be u unlabeled urns, also u colors, and b balls per color, so there 
is a total of u*b balls. 
In the following examples colors enumerated 0,1,...,u-1.
"Unlabeled" urns means that permuting urns (with their content) does not
generate new/additional countable objects.
Distribute all u*b balls until each urn is filled with b balls; the terminology
"urn" indicates that the balls of various colors inside an urn are not sorted
and considered a multiset for our purposes. If we delimit the urns 
by vertical bars and put the ball's colors in between,
these are the basic examples:

b=2, u=1 gives 1 distinct configurations:
| 0 0

b=1, u=2 gives 1 distinct configurations:
| 0 | 1

b=2, u=2 gives 2 distinct configurations:
| 0 0| 1 1
| 0 1| 0 1

b=3, u=2 gives 2 distinct configurations:
| 0 0 0| 1 1 1
| 0 0 1| 0 1 1

b=3, u=3 gives 10 distinct configurations:
| 0 0 0| 1 1 1| 2 2 2
| 0 0 0| 1 1 2| 1 2 2
| 0 0 1| 0 1 1| 2 2 2
| 0 0 1| 0 1 2| 1 2 2
| 0 0 1| 0 2 2| 1 1 2
| 0 0 2| 0 1 1| 1 2 2
| 0 0 2| 0 1 2| 1 1 2
| 0 0 2| 0 2 2| 1 1 1
| 0 1 1| 0 1 2| 0 2 2
| 0 1 2| 0 1 2| 0 1 2

b=4, u=2 gives 3 distinct configurations:
| 0 0 0 0| 1 1 1 1
| 0 0 0 1| 0 1 1 1
| 0 0 1 1| 0 0 1 1

b=5, u=2 gives 3 distinct configurations:
| 0 0 0 0 0| 1 1 1 1 1
| 0 0 0 0 1| 0 1 1 1 1
| 0 0 0 1 1| 0 0 1 1 1

b=2, u=4 gives 17 distinct configurations:
| 0 0| 1 1| 2 2| 3 3
| 0 0| 1 1| 2 3| 2 3
| 0 0| 1 2| 1 2| 3 3
| 0 0| 1 2| 1 3| 2 3
| 0 0| 1 3| 1 3| 2 2
| 0 1| 0 1| 2 2| 3 3
| 0 1| 0 1| 2 3| 2 3
| 0 1| 0 2| 1 2| 3 3
| 0 1| 0 2| 1 3| 2 3
| 0 1| 0 3| 1 2| 2 3
| 0 1| 0 3| 1 3| 2 2
| 0 2| 0 2| 1 1| 3 3
| 0 2| 0 2| 1 3| 1 3
| 0 2| 0 3| 1 1| 2 3
| 0 2| 0 3| 1 2| 1 3
| 0 3| 0 3| 1 1| 2 2
| 0 3| 0 3| 1 2| 1 2

If we put these counts into a table we have

b \ u | 1  2  3  4  5
-------------------
1     | 1  1  1  1  1
2     | 1  2  5 17 73
3     | 1  2 10
4     | 1  3
5     | 1  3

The basic question is: are my counts correct, and is this known?

Remark:
One can also rephrase the problem as considering u tables, each with b seats,
and u groups (enumerated 0,...,u-1) each with b members. In a 
sort of menage problem all u*b people are seated at the tables.
This is related to https://math.stackexchange.com/questions/1140111 :
this asks for u=16, b=3 how many configurations exist where all
tables have only people of different groups; that would be the subset
of our cases where all integers between the bars are required to be distinct.


Remark:
If we consider distinguishable/labeled urns, we end up at A257493.
The integer matrices in A257493 would indicate in rows labeled
by color how many balls of that color end up in which urn (=labeled column).

Best Regards,
Richard