More recursions:
Let f(t,m) denote the number of ways of arranging m colored balls in t slots, where t >= m and supply of each m coloured ball is infinite. Then final answer is nCmf(t,m) / nt
Here are the different recursions I have come up with for f(t,m) (alas, no direct formula in m,n)
1) mt - mC1f(t,1) - mC2f(t,2) - mC3f(t,3) .... - mCm-1f(t,m-1)
2) tC1f(t-1,m-1) + tC2f(t-2,m-1) + tC3f(t-3,m-1) + ... + tCt-m+1f(m-1,m-1)
3) mf(t-1,m-1) + m2f(t-2,m-1) + m3f(t-3,m-1) + ... + mt-m+1f(m-1,m-1)
4) m( f(t-1,m) + f(t-1,m-1) )
where f(t,1) = 1, f(t,2) = 2t-2, and f(t,t) = t!