A005119 Computation of Infinitesimal Generator

[pauldhanna at juno.com]

Seqfans,
      Thanks, Max!   
The formula in the paper is not quite correct, however; 
it is essentially given as: 
   b(n)/(n-1)! = 1/(n-1)*Sum_{i=1..n-1} C(n-i+1,i+1)*b(n-i)/(n-i-1)!
and the correct formula is: 
   b(n)/(n-1)! = -1/(n-1)*Sum_{i=1..n-1} C(n-i+1,i+1)*b(n-i)/(n-i-1)! for
n>1 with b(1)=1 
(note the missing leading sign). 
   
The sequence in the paper is a signed version of A005119; 
below I copy the formulas for unsigned terms.
  
I have submitted these formulas in a COMMENT via the OEIS submission page

and included the PARI code. 
 
Thanks,
      Paul 
  
FORMULAS for A005119 (unsigned with offset 1): 
 
a(n) = (n-2)!*Sum_{i=1..n-1} (-1)^(i+1)*C(n-i+1,i+1)*a(n-i)/(n-i-1)! for
n>1 with a(1)=1. 
 
E.g.f. satisfies: A(x) = (1-x)^2/(1-2x)*A(x-x^2) 
where A(x) = Sum_{n>=0}a(n+1)*x^n/n! with offset so that A(0)=1.
  
(PARI)
{a(n)=if(n<1,0,if(n==1,1,(n-2)!*sum(i=1,n-1,(-1)^(i+1)*binomial(n-i+1,i+1
)*a(n-i)/(n-i-1)!)))}
 

On Wed, 26 Dec 2007 22:04:20 -0800 "Max Alekseyev" <maxale at gmail.com>
writes:
> Paul,
> 
> Take a look at papers citing the one from A005119:
> http://citeseer.ist.psu.edu/context/1324051/0
> The top one there explicitly mentions A005119/M3024.
> 
> Regards,
> Max
> 
> On Dec 26, 2007 9:41 PM,  <pauldhanna at juno.com> wrote:
> > Seqfans,
> >       How does one compute the terms of A005119:
> > "Infinitesimal generator of x(x+1)."
> > 1, 1, 3, 16, 124, 1256, 15576, 226248, 3729216, 68179968, 
> 1361836800,
> > 29501349120, 693638208000, 17815908096000, 502048890201600,
> > 15388268595840000
> >
> > There is no other formula nor e.g.f. and I do not have access to 
> the
> > reference.
> > Thanks for any help,
> >       Paul
> >
> 
>