By and large, stooling is not very effective for nut trees - they're best propagated by grafting.
I've seen reports of pecan rooted by etiolating and stooling those basal suckers, but success rates have been VERY low, and I've seen no follow-up on survival rates of those few rooted specimens - it's one thing to get root formation, but another thing altogether to get the 'rooted cutting' to survive.
If it's a BW that you desire to propagate for its nut quality, you'll be much better off letting those suckers grow for a year, cutting them as scions next winter, and grafting onto seedling rootstocks.

price,
After a quick Scroogle search, I ran across a few articles on rooting pecan &/or walnut cuttings - looks like success rates with the sort of adventitious shoots you're gonna get from that BW stump will probably be better than trying to root cuttings of 'mature' wood. Still, survival and shoot elongation, post-rooting, look to be problematic.

Here's a few to get you started:

http://msucares.com/nmrec/reports/2001/fruits/etiolation.pdf

www.rngr.net/Publications/central-states-forest-tree-improvement-conference/8th-central-states-forest-tree-improvement-conference-1972/propagation-of-walnut-chestnut-and-pecan-by-rooted-cuttings/at_download/file

http://www.agriculture.purdue.edu/fnr/HTIRC/pdf/publications/Juglans.cuttings.pdf

Good luck, and let us know how it goes.