
COMMENTS

Also, for n>0, the number of totally symmetric (n1)dimensional partitions which fit in an (n1)dimensional box whose sides all have length 5.
There is no conjectured formula for a(n).
The formula a(n,d)=product{i_1=1..n, product{i_2=i_1..n, .. product{i_d=i_(d1)..n, (i_1+i_2+..i_dd+2)/(i_1+i_2+..i_dd+1)}}..} gives the number of totally symmetric ddimensional partitions that fit in a box whose sides all have length n, for d=1,2, and 3. For d>3 this formula fails. In particular, when d=4 it produces the sequence: 1, 2, 6, 32, 352, 9216, 661504... rather than the sequence above.
