Looks like you will get 50% black and 50% blue chicks. They may have some color leakage. But I don't THINK pure partridge will come out of that... maybe if bred back a second generation... but I'm not that genetically advanced to give you an accurate answer there.
I see you got the simple explanation above. Here's the info dump, in case you're interested.
What do you get when you cross an blue rooster cochin with partridge cochin hens?
Partridge is one of two colors, wild-type or pencilled. In either case, the outcome of the first generation crossing will be the same, but the genes are different.
Pencilled is a brown base (e^b/e^b) with a pencilled color pattern (Pg/Pg) and a black color (bl/bl)
Wild-type is a gold duckwing base (e+/e+) without any pencilling (pg/pg) and black color (bl/bl.)
Blue is on an extended black base (E/E) without any pencilling (pg/pg) and is actually splash and black combined (Bl/bl)
So since your rooster is blue, he'll be passing on one E gene to each and every offspring, while the hens pass on e+ or e^b to each offspring. That'll result in (E/e?) chicks. Since E is dominant, the chicks will all be a solid color, though they may have some brown leakage due to the e? gene that they will also have.
If your hens have Pg/Pg, they'll pass a copy onto their offspring, and the rooster will pass on a pg. That'll result in Pg/pg, which will be pencilled, but not a very nice pencilling (there need to be two copies (Pg/Pg) for proper expression.) The pencilling won't show on a black or blue background, but it might show in the second generation. If the hens don't have pencilling, the offspring will be pg/pg.
Which leaves us with blue. Since the rooster is (Bl/bl), he can pass on splash (Bl) or black (bl) to his offspring, which will inherit black (bl) from their mothers. So they'll be either (Bl/bl) (blue) or (bl/bl) (black)
Is there some % possibility to get partridge chicks? Nope. Partridge is recessive to Extended black, which is the color pattern on which blue is based.
Partridge Cochin are the penciled type partridge.
Silver penciled are the same except silver based while partridge is gold based.
You can use silver penciled hens with a partridge rooster and produce partridge pullets. Cockerels won't be pure partridge in the first generation though.