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Let $S$ be a set of objects, and suppose $w$ is a weight function on $S$ with generating function $\Phi_S(x)$. Let $w^*$ be a new weight function for $S$ defined by $w^*(a)=5w(a)+3$ for all $a\in S$, and let $\Phi^*_S(x)$ be the generating function for $S$ with respect to $w^*$. Express $\Phi^*_S(x)$ in terms of $\Phi_S(x)$.

How would I do this problem? Is there some general method for linearly changing the weight function? Google seems to be extremely unhelpful with my introductory combinatorics course.

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This is $x^3 \Phi_S(x^5)$ by inspection. –  Marko Riedel May 20 '13 at 21:56
    
How did you inspect it? ;) –  user54609 May 22 '13 at 19:25
    
Well if you take just one term of $\Phi$ it is given by $x^{w(a)}$. So to get $5w(a)$ use $(x^5)^{w(a)}$ and for $3+5w(a)$ use $x^3(x^5)^{w(a)}.$ –  Marko Riedel May 22 '13 at 19:46
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