# [seqfan] Re: A family of quadratic recurrences

A.N.W.Hone A.N.W.Hone at kent.ac.uk
Mon Oct 5 14:22:17 CEST 2009

```Dear Richard,

The sequences that Jaume considered cannot be generated by recurrences of the type you mention, because
they have double exponential growth, i.e.

log log a_n \sim  K n

for some positive constant K (which I can give explicitly for Jaume's examples, if anyone is interested).

The growth of these sequences is too fast for them to satisfy linear recurrences of the type you mention.
In fact, the  asymptotics is found from a linear recurrence for log a_n which is only satisfied to leading order;
so I doubt that even log a_n has a nice generating function.

Andy

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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 [1] 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

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