A Function With Integer Derivatives

[Leroy Quet]

Consider the function, defined for -1 < x < 1,

f_m,n(x) =

(1-x)^((1-x)^(-m)) (1+x)^((1+x)^(-n)),

where m and n are positive integers.

Now, the derivatives of f at x = 0 are all integers.

What I am interested in is the array, {a(m,n)}, where
a(m,n) = the (m+n+1)th order derivative of f_m,n(x) at x = 0.

The reason I am interested in this derivative is that
a(m,n) is always divisible by (m+n).

(Actually, more generally, the (j+1)th order derivative of f_m,n(x) is 
always divisible by
GCD(m+n,j).)

So, could someone please calculate/submit the sequence of {a(m,n)} (read 
off by diagonals) and also submit {b(m,n)}, where b(m,n) = a(m,n)/(m+n)?

Is there a direct non-recursive way of calculating {a(m,n)} and {b(m,n)} 
without using calculus?
(I have not actually calculated the terms myself, so this may be trivial 
to impossible.)


thanks much,
Leroy Quet