co.combinatorics - Partition-Like Counting

Fix $k,ngeq 1$. I wish to know the number of ways of dividing $n$ up as follows. First find $c_1,c_2,ldots ,c_k$ such that $c_igeq 0$ and $sum_ic_i=n$. Then take each $c_i$ and find $c_{i,1},ldots ,c_{i,n}$ such that $c_{i,m}geq 0$ and $sum_mmc_{i,m}=c_i$. How many ways are there to do this?