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I will be very happy to understand how to solve this problem with generating function: How many solutions are there to the equation $$x_1 + x_2 + x_3 + x_4 + x_5 = 31$$ where $x_i$ is a nonnegative integer, and $x_2$ is even and $x_3$ is odd?

thanks!

Matthew Conroy
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sapir
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2 Answers2

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Non-negative $x_1$ corresponds to $\frac{z}{1 - z}$, $x_2$ even gives rise to $\frac{1}{1 - z^2}$, odd $x_3$ means $\frac{z}{1 - z^2}$, non-restriced $x_4$, $x_5$ are $\frac{1}{1 - z}$ each: \begin{align} [z^{31}] &\frac{z}{1 - z} \cdot \frac{1}{1 - z^2} \cdot \frac{z}{1 - z^2} \cdot \frac{1}{(1 - z)^2} \\ &= [z^{31}] \frac{z^2}{(1 - z)^3 (1 - z^2)^2} \\ &= [z^{29}] \left( \frac{5}{64 (1 + z)} + \frac{1}{32 (1 + z)^2} + \frac{5}{64 (1 - z)} + \frac{1}{8 (1 - z)^2} + \frac{3}{16 (1 - z)^3} + \frac{1}{4 (1 - z)^4} + \frac{1}{4 (1 - z)^5} \right) \end{align} Use the fact that $\binom{-n}{k} = (-1)^k \binom{k + n - 1}{n - 1}$: \begin{align} [z^{31}] &\frac{z^2}{1 - z} \cdot \frac{1}{1 - z^2} \cdot \frac{z}{1 - z^2} \cdot \frac{1}{(1 - z)^2} \\ &= \frac{5}{64} \binom{-1}{29} + \frac{1}{32} \binom{-2}{29} + \frac{5}{64} \binom{-1}{29} (-1)^{29} + \frac{1}{8} \binom{-2}{29} (-1)^{29} + \frac{3}{16} \binom{-3}{29} (-1)^{29} + \frac{1}{4} \binom{-4}{29} (-1)^{29} + \frac{1}{4} \binom{-5}{29} (-1)^{29} \\ &= 11560 \end{align}

vonbrand
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  • thank you very much :) but one thing I don't understand.. why x1 is diffrent from x4 and x5? they all have the same conditions – sapir Apr 19 '14 at 22:15
  • @sapir, sorry. It seems I misread the condition on $x_i$ as a condition on $x_1$. I solved it assuming $x_1 \ge 1$, $x_4, x_5 \ge 0$. – vonbrand Apr 19 '14 at 22:53
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The generating function is $(1/(1-x))^3 \times 1/(1-x^2) \times x/(1-x^2)$. (Why?) The coefficient of $x^{31}$ in the series expansion gives you your answer.

user2566092
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