Well, back in NJ, we had a zillion red-tailed hawks, a group of about 6 turkey vultures, a golden eagle pair, googleplex of crows, but we were lacking jays there, too, now that you mention it. There were a few. More cardinals than jays, though, which now that you mention it was quite odd. (Yes, I know Golden Eagles are not an east-coast species, but there they were. They are sorta hard to miss sitting in the maple tree in the front yard, let me tell you! I think the tree was starting to lean, though.

)
We also had visiting grossbeaks spring and fall, a googleplex of assorted little songbirds, and every once in a while an invasion of rooks.
But in 2002 on the drive across, there were lots of Blue Jays in Ohio, at least.
The crows, I'm told, are taking it on the chin in NJ now.
Here in Washington we have a FEW, a small number of songbirds, a bunch of Ravens (I thought crows were a pain!), and (yes, I'm not kidding) a (*&(* Big Eared Owl living in a dead tree in the backyard (I should explain, my backyard and my neighbors backyards make up about 5 acres of original cedar forest, about 125 to 150' high, including deadfalls and snags)
Flippin owl, well, it's nice to have around, but when it perches on the skylight (that would be bedroom skylight, 3am

) and starts to hoot, I swear I'm going to start tossing slippers at it.
But we haven't many mice, and not a rat, and the squirrels are the most speedy, paranoid squirrels I've seen in a long time. You just don't want to walk on the ground underneath its nest is all, EWWW!