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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
 #1
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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

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