If we look at the difference between the number of partitions of $n$ with distinct parts that have an even number of parts and the number of partitions of $n$ with distinct parts that have an odd number of parts we get Euler's pentagonal theorem. What happens when we look at the difference between the number of partitions of $n$ with parts differing by at least 2 that have an even number of parts and the number of partitions of $n$ parts differing by at least 2 that have an odd number of parts. Is this difference limited some way? Is it ever bigger in absolute value than one? Are there known upper or lower bounds as n goes to infinity?

1$\begingroup$ It is 2 for $n=8$ :) $\endgroup$ – მამუკა ჯიბლაძე Mar 15 '16 at 18:53

1$\begingroup$ Related: Rogers' Theorem, at en.wikipedia.org/wiki/Glaisher%27s_theorem $\endgroup$ – Gerry Myerson Mar 15 '16 at 22:06
The graph certainly looks promising :)
As for the generating function  what I managed to get is a very strange "downsideup continued fraction" $$ 1qq^2\frac{1q^2q^4\frac{1q^3q^6 \frac{1q^4q^8\frac{ 1q^5q^{10}\frac{1q^6q^{12}\cdots}{1q^5}}{1q^4}}{1q^3}}{1q^2}}{1q} $$ Looks like it is A039924, in which case this generating function is equal to $$ 1\frac q{1q}+\frac{q^4}{(1q) \left(1q^2\right)}\frac{q^9}{(1q) \left(1q^2\right) \left(1q^3\right)}+...$$

$\begingroup$ I assume this is a graph of the difference in question. How was it computed? Could it be extended further or are there computational obstacles preventing this after 50. It raises some more questions: Does this graph ever stabilize at one sign? Is its increase or decrease as n goes to n+1 limited by some integer? What is that integer if it exists? Can you show the absolute value exceeds any integer? $\endgroup$ – Kristal Cantwell Mar 15 '16 at 19:07

$\begingroup$ I used quick and dirty Mathematica code
ListPlot[Table[With[{l = Select[IntegerPartitions[n], Max[Differences[#]] < 1 &]}, 2Length[Select[l, EvenQ@*Length]]  Length[l]], {n, 50}], Joined > True]
, surely it can be done much better $\endgroup$ – მამუკა ჯიბლაძე Mar 15 '16 at 21:58 
$\begingroup$ @KristalCantwell Thanks for accepting but this does not really answer any of your questions :D So I will leave it cw I think... $\endgroup$ – მამუკა ჯიბლაძე Mar 17 '16 at 6:32

$\begingroup$ It was useful especially the link to A039924. $\endgroup$ – Kristal Cantwell Mar 17 '16 at 17:00