Rich's proposal is the b=a/2 case of
In[722]:= TableForm[Table[
Limit[Binomial[a*t + n, b*t + k], t -> 0], {n, -3, 3}, {k, -3, 3}]]
Out[722]//TableForm=
b b 3 b 6 b 10 b
- 1 - - -3 + --- 6 - --- -10 + ----
a 0 0 a a a a
2 b b b 2 b 3 b 4 b
-(---) - 1 - - -2 + --- 3 - --- -4 + ---
a a 0 a a a a
b b b b b b b
- -(-) - 1 - - -1 + - 1 - - -1 + -
a a a a a a a
0 0 0 1 0 0 0
0 0 0 1 1 0 0
0 0 0 1 2 1 0
0 0 0 1 3 3 1
In[723]:= % ==
Table[Limit[Binomial[a*t + n, a*t + n - b*t - k], t -> 0], {n, -3, 3}, {k,
-3, 3}] ==
Table[Limit[(a*t + n)!/(b*t + k)!/(a*t + n - b*t - k)!, t -> 0], {n, -3,
3}, {k, -3, 3}]
Out[723]= True
GKP says b=0*a.
Kronenburg and Mma say that 97% of mathematicians agree that the science is
settled.
--rwg
