[1] It is most transparent when stated in its most general form, for joint cumulants, rather than for cumulants of a specified order for just one random variable.
The case n = 2 is well-known (see law of total variance).
For general 4th-order cumulants, the rule gives a sum of 15 terms, as follows: Suppose Y has a Poisson distribution with expected value λ, and X is the sum of Y copies of W that are independent of each other and of Y.
We have: We recognize the last sum as the sum over all partitions of the set { 1, 2, 3, 4 }, of the product over all blocks of the partition, of cumulants of W of order equal to the size of the block.
That is precisely the 4th raw moment of W (see cumulant for a more leisurely discussion of this fact).