An Unusual Generating Function

[Leroy Quet]

[also posted to sci.math]


Let C(m) be the mth Catalan number,
C(m) = binomial(2m,m)/(1+m).)

Let S(m,n) = an unsigned Stirling number of the first kind.
(S(0,0)=1, S(0,n)=0 for n not 0,
S(m+1,n) = m*S(m,n) + s(m,n-1).)
 

OK, let

a(m) =

sum{k=0 to m} S(m,k) C(k) (-1)^(k+m).


(If I did not error figuring the first few terms by hand,
a(m) -> 1, 1, 1, 1, 0, 1,...)


Let f(y) =

sum{m=0 to oo} a(m) y^m /m!.


I believe that f(y) =

1 + integral{0 to y} f(x) f((y-x)/(1+x))/(1+x) dx,

which is in ascii-art mode:

f(y) =
          
     /y         y-x      dx
1 +  |  f(x) f( --- )  -----
    /0          1+x    (1+x)
     

(unless I erred)
 

But what is a closed-form for {a(m)} and/or f(y) ??

thanks, 
Leroy
      Quet