# [seqfan] Re: A family of quadratic recurrences

Richard Mathar mathar at strw.leidenuniv.nl
Sun Oct 4 15:03:08 CEST 2009

```Starting from http://list.seqfan.eu/pipermail/seqfan/2009-September/002400.html

> 
> A059480 A recurrence equation.
> a(n) = a(n - 1) + (n + 1)*a(n - 2)
>
> a[i+1]=( a[i-2]*a[i-1]-a[i-1]^2+a[i-2]*a[i]+a[i-1]*a[i])/a[i-2]
> (i think i proved this by substituting the definition and simplifying to
> 0)

> In the family of quadratic recurrences defined by
> a(n)=sum(i=1,L-1,a(n-i)*sum(j=i,L-1,a(n-j)))/a(n-L), with L initial ones,
> I have not been able to find any noninteger value.
>
> Do these recurrences yield only integers? For any L>=2?
>
> This search is related to sequence
> http://research.att.com/~njas/sequences/A165896 ,
> which is the case L=4.
>

I wonder whether all these sequences defined by convoluted products can be
written with simpler recurrences of the polynomial format
a(n) = sum_j (some polynomial in n, depending on j)* a(n-j).
So, can one reverse the type of operation indicated in the first message?

As a side note, we observe that equations like  map onto 1st order
differential equations for their generating function (where 1st means that the
polynomials in front of the a(n-j) on the right hand side no higher than 1st order).
So it would be nice to solve these to produce the generating function.
(Would the Zeilberger telescoping
http://www.math.rutgers.edu/~zeilberg/programs.html
http://mathworld.wolfram.com/ZeilbergersAlgorithm.html
http://mathworld.wolfram.com/Wilf-ZeilbergerPair.html
help to solve this?)

Richard

```