An Unusual Generating Function

[Leroy Quet]

'Vladeta' has written me that {a(m)} is the EIS-sequence A086672.

There it gives f(y) as:

>E.g.f.: hypergeom([1/2],[2],4*ln(1+x)) =
> (1+x)^2*(BesselI(0,2*ln(1+x))-BesselI(1,2*ln(1+x)))"


thanks,

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
>