[seqfan] Re: A family of quadratic recurrences
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
and reading http://list.seqfan.eu/pipermail/seqfan/2009-September/002455.html
> 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
help to solve this?)
More information about the SeqFan