Variant of binomial coefficients I've recently come across a variant of the binomial polynomials, and I'm curious if anyone has seen these before.  If so, I'd love a reference, a name, etc.
First recall the following.  If z is a formal variable, then we can consider \binom{z}{k} as a polynomial in z by the standard formula: \binom{z}{k}= [z(z-1)...(z-k+1))]*[k!]-1.
Here's the variant I came across.  Let a and k integers, where a divides k, and we write k=ab.  Consider the polynomial 
F(a,k)(z)=[z(z-a)(z-2a)...(z-k+a))][k(k-a)(k-2a)...(a)]-1.
Has anyone this before?  Or anything similar?
Update:  Jonah helpfully identifies a typo in the numerator of F(a,k)(z) which I've now corrected.
 A: On the other hand, maybe what you want is a combinatorial interpretation.  There is a way to express a related function as a binomial determinant which can be combinatorially interpreted using Gessel-Viennot, as described here.
A: (For simplicity, you probably want the last term in the numerator to be z-k+a, right? That way F(1,k)(z)=\binom{z}{k}. I'll pretend that's what you meant.)
I haven't come across such polynomials, but they're easily expressed in terms of multifactorials. Namely,
F(a,k)(z)=z!(a)/[(z-k)!(a)k!(a)]
Note that this isn't an integer when a doesn't divide z. EDIT: (And, as Qiaochu points out, when a does divide z it's just regular ol' z-choose-k.)
A: Let Bk(z) denote the usual binomial coefficient, suitably generalized to allow z to be a formal variable.  Assuming that you intend to have the same number of factors in the numerator and denominator in your definition of F(a,k)(z), then via the substitution z = aw,
F(a,ab)(z) = Bb(w).
If 1/a is in your ring (still assuming that a divides k), then
F(a,k)(z) = Bk/a(z/a).
