2 added 872 characters in body

Chris Phan's comment sounds right to me, but you may be able to do this more quickly, though OEIS doesn't seem to have any details on rows or columns of the below array except that they're (at least often) multinomial coefficients. I have computed these numbers in MATLAB using this stuff:

for n=1:8,for k=1:9,L=lookup(k,n);L1(k,n)=sum(sum((L==1),2)>0);end,end,L1

L1 =

       1           0           0           0           0           0           0           0
2           1           2           2           2           2           2           2
3           3           7           9          12          15          18          21
4           6          16          25          40          58          80         106
5          10          30          55         101         165         255         375
6          15          50         105         216         391         666        1071
7          21          77         182         413         819        1520        2646
8          28         112         294         728        1568        3144        5881
9          36         156         450        1206        2802        6030       12051


As an example, consider k=4 and n=3: MATLAB gives (I have added asterices for clarity)

lookup(4,3)

ans =

 0     0     0     3
0     0     1     2     *
0     0     2     1     *
0     0     3     0
0     1     0     2     *
0     1     1     1     *
0     1     2     0     *
0     2     0     1     *
0     2     1     0     *
0     3     0     0
1     0     0     2     *
1     0     1     1     *
1     0     2     0     *
1     1     0     1     *
1     1     1     0     *
1     2     0     0     *
2     0     0     1     *
2     0     1     0     *
2     1     0     0     *
3     0     0     0


and visual inspection shows that the number of rows with at least one unit entry is 16, identical with the table entry.

1

Chris Phan's comment sounds right to me, but you may be able to do this more quickly, though OEIS doesn't seem to have any details on rows or columns of the below array except that they're (at least often) multinomial coefficients. I have computed these numbers in MATLAB using this stuff:

for n=1:8,for k=1:9,L=lookup(k,n);L1(k,n)=sum(sum((L==1),2)>0);end,end,L1

L1 =

       1           0           0           0           0           0           0           0
2           1           2           2           2           2           2           2
3           3           7           9          12          15          18          21
4           6          16          25          40          58          80         106
5          10          30          55         101         165         255         375
6          15          50         105         216         391         666        1071
7          21          77         182         413         819        1520        2646
8          28         112         294         728        1568        3144        5881
9          36         156         450        1206        2802        6030       12051