(Okay, I'll confess: I knew the first term, and I got Maple to calculate the others just now.)
So I found myself wondering -- why this n-3/2? Let Dn = 4-n Cn. Then
and so we get
; furthermore that sum is about -(3/2) log n, for large n, and so Dn is about n-3/2. The replacement of 1-x with exp(-x) obviously would need to be justified (and such justification would explain the presence of the mysterious π) but I'm still amused that this simple computation got the exponent.