Let $P(x)=\sum_{n=0}^{\infty} p_nx^n$ be the partition generating function, and let $P^*(x)=\sum_{n=0}^{\infty} p^*_nx^n$, where $$p^*_n = \binom{\text{number of partitions of }n}{\text{into an even number of parts}} - \binom{\text{number of partitions of }n}{\text{into an odd number of parts}}.$$ For example, $p^*_4=3-2=1$, because there are $3$ partitions of $4$ into an even number of parts $(3+1,\ 2+2,\ 1+1+1+1)$ and $2$ partitions of $4$ into an odd number of parts $(4,\ 2+1+1)$.

Compute the truncation of $P(x)P^*(x)$ to degree $10$; that is, determine the polynomial consisting of all terms in the power series expansion of $P(x)P^*(x)$ with degree less than or equal to $10$.

(As an example, the truncation of $\frac 1{1-x}$ to degree $3$ is $1+x+x^2+x^3$.)


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