are partitions of n into perfect powers the same as Molien series?

[Marc LeBrun]

I just submitted the table A102430 (and some related sequences) defined by
T(n,k) = partitions of n into k raised to non-negative powers < n:

       1  1 1 1 1 1 1 1 1 1 1 1
       3  2 1 1 1 1 1 1 1 1 1 1
      10  2 2 1 1 1 1 1 1 1 1 1
      35  4 2 2 1 1 1 1 1 1 1 1
     126  4 2 2 2 1 1 1 1 1 1 1
     462  6 3 2 2 2 1 1 1 1 1 1
    1716  6 3 2 2 2 2 1 1 1 1 1
    6435 10 3 3 2 2 2 2 1 1 1 1
   24310 10 5 3 2 2 2 2 2 1 1 1
   92378 14 5 3 3 2 2 2 2 2 1 1
  352716 14 5 3 3 2 2 2 2 2 2 1
1352078 20 7 4 3 3 2 2 2 2 2 2

For example T(9,3) are the 5 partitions 111111111, 1111113, 1113, 333 and 9.

Column 1 is choose(2n-1,n) = A001700 and column k>1 obeys
T(n,k) = T(n-1,k) + (T(n/k,k) if k divides n, else 0)

Column 2 is given by A018819, and Column 3 by A062051.

However, Columns 2 through 5 *also* seem to be the same as the "Molien 
series" entries A008645, A008650, A008652 and A008648.

Are these in fact the same?  If so, can someone provide an elementary 
explanation of Molien series (of which there are a zillion)?

Thanks!