My first intuition on how to solve this problem would be to use a computer program to calculate all the ways the items can be distributed for n items and finding out how many of the ways have bins have bins with k or more items.

For instance, if n = 5 the items can be distributed as follows:

1+1+1+1+1, 2+1+1+1, 3+1+1, 2+2+1, 3+2, 4+1, 5

Then, for each of these partitions find out how many ways they can be spread out over m bins. However, this becomes infeasible quickly because for n=100 there are 190569292 possible partitions and for n=1000 there are 24061467864032622473692149727991 partitions.