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This is due today, so help fast! 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  and 2 partitions of 4 into an odd number of parts .

Compute the truncation of  to degree $$P(x)P^*(x)$$ ; 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 $$1+x+x^2+x^3$$.)

Jul 10, 2020

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Using a 1-1 correspondence, we can show that $$p_n^* = p_n - p_{n - 1}$$ (*) for all n >= 2.  So,

$$P^{*}(x) = 1 - x - x^3 + x^4 - x^5 + \dotsb$$

The degree 10 truncation of $$P(x) P^{*}(x)$$ is then

$$1 + x^2 + x^4 - 2x^5 - x^6 + x^8 - 3x^9 + x^{10}$$

(*) "Theory of Partitions", George E. Andrews

Jul 10, 2020