# [seqfan] Even-odd partition function dissection (A058695 + A058696)

William Keith william.keith at gmail.com
Sat Feb 15 20:58:32 CET 2014

In A058695 + A058696, the number of partitions of 2n+1 and of 2n
respectively, there are two comments by Michael Somos that together give an
even-odd dissection of the partition function, i.e.

1/(q;q)_\infty = (q^12,q^16,q^20,q^32;q^32)_\infty/((q^2;q^2)^2_\infty
(q^6,q^10;q^16)_\infty) + q (q^4,q^16,q^28,q^32;q^32)/((q^2;q^2)^2_\infty
(q^2,q^14;q^16)_\infty)

or, using the easier-to-read notation (q^a,...,q^b;q^c)_\infty =:
(a,...,b;c),

1/(1;1) =
(12,20;32)(16;16)/((2;2)^2(6,10;16))
+ q(4,28;32)(16;16)/((2;2)^2(2,14;16)).

He states these identities without proof, and I was wondering if anyone
knew of such a proof.  I have dropped a line on his Wiki user discussion
page, but the comments are over 10 years old and his user page has had no
activity so I was not confident of contacting him that way.

These identities would be useful to me in a paper, and so I would need a
solid cite or a quotable proof.

Thanks,
William Keith