Find coefficient of $x^{50}$ in $(\sum_{i=1}^{\infty}x^n)^3$

Question

How do you find the coefficient of $x^{50}$ in $(\sum_{i=1}^{\infty}x^n)^3$?

The sum inside the brackets is $\frac{1}{1-x}$. Then the number you are looking for, using Taylor, is $f^{(50)}(0)/50!$ for $f(x)=(1-x)^{-3}$. — OR., Oct 03 '13 at 07:29

score 4 · Answer 1 · answered Oct 03 '13 at 07:31

4

Hint: This is (for $|x|\lt 1$) equal to $\frac{x^3}{(1-x)^3}$. Do you know the expansion of $(1-x)^{-3}$ (general binomial theorem)? We want the coefficient of $x^{47}$.

answered Oct 03 '13 at 07:31

André Nicolas

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how do you apply binomial theorem to a negative power? – James Lee Oct 03 '13 at 07:38
You might want to look here.. You may also have done the result in class as an application of "Stars and Bars." – André Nicolas Oct 03 '13 at 07:51

score 4 · Answer 2 · answered Oct 03 '13 at 07:41

Another way to look at this is purely combinatorial: the number of integer solutions to: $$a_1+a_2+a_3=50;\quad a_1,a_2,a_3>0.$$ (Or, equivalently, the number of ways to put 50 balls into 3 urns such that no urn is left empty.)

That number is well-known to be $\binom{50-1}{3-1}$.

Find coefficient of $x^{50}$ in $(\sum_{i=1}^{\infty}x^n)^3$

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