Each term is close to (m- k+ sqrt(d) + 1/2)(m - k- sqrt(d) + 1/2), where k ranges from 1 to h=(m-n)/2. (h may be off by 1.) Your product is close to n! squared times the product of two binomial coefficients: m+ a choose h and m- b choose h, where a anb are close to sqrt(d). Letting a and b range over floor(sqrt(d)) and ceil(sqrt(d)) are a start if h is not too small. For finer estimates, let a and b be appropriate real numbers.
Gerhard "Email Me About System Design" Paseman, 2011.06.21