Recurrence relation for Stirling number of first kind with a minimum value

As it states in Wikipedia: Stirling numbers of the first kind count permutations according to their number of cycles. And the recurrence relation is as follows:

$$left({n+1atop k}right) = n left({natop k}right) + left({natop k-1}right)$$

I was wondering how the recurrence relation will look like if there was a lower bound on the size of the cycle ( at least x nodes in a cycle ).