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I want to find the coefficient of $x^{10}$ in the expansion:

$(1 + x + x^2 + ... + x^{10})(1+x^2 + x^4 + ... + x^{10})(1+ x^5 + x^{10})$

I deduced this problem to:

$\sum\limits_{i=0}^{10}\sum\limits_{j=0}^{5}\sum\limits_{k=0}^{2}x^{i + 2j + 5k}$

So I have to find the solutions to:

$i + 2j + 5k = 10$ where $i,j,k$ are integers. How do I do this other than guessing solutions? Or is there another way to deduce the coefficient of $x^{10}$ in this problem?

2 Answers 2

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this is equal to the solutions to $2j+5k\leq 10$.

There is clearly one solution when $k=2$, three solutions when $k=1$ and six solutions when $k=2$.

So the answer is $10$.

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Extend the finite series to infinite series and compute the exponent of $x^{10}$ in the product (this does not change the answer). Hence it suffices to find the coefficent in the expansion of $$ \frac{1}{(1-x)(1-2x)(1-5x)} $$ which you can compute using partial fractions.

  • 0
    Is it possible that you made a mistake in your denominators? – 2017-01-06
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    I think it should be $(1-x)(1-x^2)(1-x^5)$ – 2017-01-06
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    Yes I think so too. Computing partial fractions on these seems too much work. – 2017-01-06