Help needed with new sequences

[wouter meeussen]

sorry if I don't get it, but
since
Sum[x^(k(k + 1)/2), {k, 0, \[Infinity]}]
=EllipticTheta[2, 0, x]/(2 x^(1/8))

and
Sum[x^(k*k), {k, 0, \[Infinity]}]
=1/2*(1 + EllipticTheta[3, 0, x])

it is easy to write the GF for this:

I find
CoefficientList[Series[EllipticTheta[2, 0, Sqrt[x]]/(2*x^(1/8))*1/2*
    (1 + EllipticTheta[3, 0, x])*1/2*(1 + EllipticTheta[3, 0, x]), {x, 0, 64}], x]
=
{1, 3, 3, 2, 4, 5, 3, 4, 4, 3, 7, 7, 3, 4, 5, 5, 8, 5, 4, 9, 8, 4, 4, 8, 3,
8, 12, 4, 8, 7, 7, 8, 9, 4, 4, 11, 5, 10, 7, 5, 10, 10, 5, 4, 8, 4, 11, 7, 4,
4, 9, 9, 6, 12, 4, 10, 7, 4, 6, 6, 6, 6, 8, 2, 10}

Am I being silly here?

W.

----- Original Message ----- 
From: "N. J. A. Sloane" <njas at research.att.com>
To: <seqfan at ext.jussieu.fr>
Cc: <njas at research.att.com>
Sent: Saturday, May 14, 2005 1:33 AM
Subject: Help needed with new sequences



Dear Seqfans,  Professor Zhi-Wei Sun and his students have been looking
at numbers that can be written (for example) in the form x^2 + y^2 + T_z,
and they show that every natural number n can be written in this form.
(T_z = z(z+1)/2)

So it is natural to ask, how many ways are there of writing
n in the form x^2 + y^2 + T_z?  (S+S+T, for short)

They consider MANY other mixed sums of squares (S) and triangular numbers (T)
(such as S+2S+T, S+2T+T, ...)

There are two papers on the arXiv:
http://front.math.ucdavis.edu/math.NT/0505128
http://front.math.ucdavis.edu/math.NT/0505187
and possibly more on his home page:
http://pweb.nju.edu.cn/zwsun

For each of these we may ask, how many ways are possible?
And there can be several answers, depending on
whether order or signs are taken into account.

Sequence A005875 is the classical sequence that gives the number
of ways of writing n as a sum of 3 squares, taking
order and signs into account.
But if you ignore signs and order you
get a sequence which begins (I think) 1,1,1,1,1,1,0,1,2,1,...,
-I think David Wilson would call the latter sequence
"Number of ways to partition n into 3 or fewer squares"
(I didn't stop to find out the A-number).

So here are a lot of potential new sequences from Sun's papers,
if one or more seqfans would like to compute them!
There is enough material for a collaborative effort.
If you are going to work on this, post a note here.  Thanks!
NJAS