Q: relation A059348 and A104722

[sellersj at math.psu.edu]

All,

It appears that the generating functions for the two sequences are the
same (ignoring the constant terms which is the current difference between
the two sequences).  Here is a proof, although not necessarily the most
elegant.

First, the generating function given in A104722 is

(1+x)^2*C(x^2)^2 where C(x) = (1-sqrt(1-4x))/(2x) is the generating
function for the Catalan numbers.  When this is expanded, and we ignore
the constant term, we obtain

( 1 + 2x - x^2 - 4x^3 - (x+1)^2sqrt(1-4x^2) )/(2x^4).

Next, consider the construction of the values of A059348.  By looking at
page 99 of the reference given in the OEIS entry (which is F. R. Bernhart,
Catalan, Motzkin, and Riordan numbers, Discr. Math., 204 (1999) 73-112),
it is apparent that these entries are built as follows:

First, take the sequence of "aerated" Catalan numbers (the Catalans padded
with zeros):

1 0 1 0 2 0 5 0 14 0 42 0 132 ...

This sequence has generating function C(x^2).  Then take the sum of each
pair of consecutive values, like one might do to build the next row of
Pascal's triangle from the previous row, to yield the sequence

1 1 1 2 2 5 5 14 14 42 42 132 132 ....

The generating function for this auxiliary sequence is given by

((1+x)C(x^2) - 1)/x .

We then sum pairs of consecutive values once more to obtain

2 2 3 4 7 10 19 28 56 ...

which is the sequence of values in A059348.  So this generating function is

(1+x)[((1+x)C(x^2) - 1)/x -1] / x

again ignoring the constant term.

Via straightforward algebraic manipulations, one can show that this
quantity is also equal to

( 1 + 2x - x^2 - 4x^3 - (x+1)^2sqrt(1-4x^2) )/(2x^4)

(with the reminder that we are ignoring the constant term).

So this proves that the sequences are the same if we ignore the initial
term in both.

I am confident that a prettier proof exists, but this one is still
satisfying.

All the best!

James Sellers



> Almost certainly.   Both have rational generating functions.
>
> Franklin T. Adams-Watters
>
> -----Original Message-----
> From: Richard Mathar <mathar at strw.leidenuniv.nl>
>
> The next question that arises in search of duplicates is:
> Are the two sequences
> http://research.att.com/~njas/sequences/?q=id:A104722|id:A59348
> the same apart from the initial term?
>
> Richard
>