I'm studying chapter 14 "Partitions" of the famous Apostol's Introduction to Analytic Number Theory. Down at page 311 (section 14.4) and endeavoring to study the pentagonal numbers, Apostol writes on pentagonal numbers:
... are also the partial sums of the terms in the arithmetic progression 1,4,7,10,13,...,3n+1,...
If $\omega(n)$ denotes the sum o the first n terms of this progression then $\omega(n)=\frac{3n^2-n}{2}$.
Then he defines the pentagonal numbers as being the number $\omega(n)$ and $\omega(-n)=\frac{3n^2+n}{2}$.
I don't get what $\omega(-n)$ here represents, I need help understanding the context of this value and its implications on the definition of Pentagonal numbers.
Following up on jjagmath's comment, putting negative arguments into the function $\omega$ abandons the interpretation of partial sums of $3n+1$. Yet these "generalized pentagonal numbers" are essential for Euler's recurrence for the partition numbers.
This is an example of a phenomenon called combinatorial reciprocity: Given a function $f(n)$ that counts a combinatorial object based on a parameter $n \ge 0$, there can be something else that is counted by $f(-n)$ (or $-f(-n)$). Beck and Sanyal's book Combinatorial Reciprocity Theorems (AMS 2018) is a nice exploration of these ideas.
Back to this particular problem. As it happens, one can show that $\omega(-n) = (3n^2+n)/2$ is a different summation: $$ \omega(-n) = \sum_{j=1}^n n+j.$$ For example, $n = 3$ gives $4 + 5 + 6 = 15$, then $n=4$ gives $5+6+7+8 = 26$.
This connects to the "geometric" pentagonal numbers in a similar way. You can show that $$\omega(n) = \sum_{j=0}^{n-1} n + j,$$ e.g., $3+4+5 = 12$ and $4 + 5 + 6 + 7 = 22$.