Lily, Magic Lily, Resurrection Lily, Naked Lady

I have 2 huge clumps of this lily by my walk. Right now there are many leaf fans, yet during the past couple of years I have noticed that only a few blooms appear. How do I encourage it to put up more flower stalks?
When would I correctly be able to divide the clumps?

I like to divide them as the foliage is dying off in late spring because I can see where all the bulbs are. Crowding could be the reason for fewer blooms, but too much shade encroaching upon them can cause this too. If shade is not an issue, I'd divide them this spring and replant with some bulb food (I like bone meal), and perhaps amend the soil with compost if its lean.

I've noticed that it takes a few years after planting before you'll get a lot of blooms again, once you divide. Leaves a-plenty, but it takes a while for flower production to catch up again.