$$
f(n) = \frac{n(3n-1)}{2} \, (n \in (-\infty, +\infty))\
P(n) = (1 + x + x^2 +…)(1+x^2+x^4+…)(1+x^3+x^6+…)…..\
= \frac{1}{(1-x)(1-x^2)(1-x^3)….}\
\
\phi(x) = \prod_{i=1}^{\infty}(1-x^{i})
= (1 - x - x^2 + x^5 + x^7 - x^{12}…)
= \sum_{k=-\infty}^{\infty}(-1)^{k}x^{\frac{k(3k-1)}{2}}
= \sum_{k=0}^{\infty}(-1)^{k}x^{\frac{k(3k \pm 1)}{2}}\
(1-x-x^2+x^5+x^7-x^{12}…)(1 + p(1)x + p(2)x^2 + …) = 1\
p(n) = p(n-1)+p(n-2)-p(n-5)-p(n-7)+p(n-12)..\
$$