# Determine the coefficient of expansion of the product of two sumations?

I would like to determine the coefficient of a desired term in the product of two summation where the powers of $x$ are not necessarily integers. For example $$\sum_{i=1}^N x^{1/i}\sum_{j=1}^N x^{1/j}$$ where $N=50$ is given. I want to know what is the coefficient of $x^{3/10}$ where I calculated just by $\texttt{Exp[ ]}$ command which is 4. Is there any direct and faster way to calculate such coefficients without expanding the product (as if $N$ increases, it would take a long time to expand it and also difficult to find the desired coefficient).

Thanks!

prod[n_Integer?Positive] := Sum[x^(1/i), {i, n}]*
Sum[x^(1/i), {i, n}];

Coefficient[prod, x^(3/10)]


4

Or

Coefficient[prod, x, 3/10]


4

You might try this:

FindInstance[{j > 0, i > 0, 1/j + 1/i == 3/10}, {j, i}, Integers, 10]


This paper on page 19 proves that there are only 4 (counting permutations) Egyptian fractions for 3 / 10.

http://www.nntdm.net/papers/nntdm-19/NNTDM-19-2-15-25.pdf

So no matter how large N is 4 is the maximum value of that coefficient.

• Or Reduce[{j > 0, k > 0, 1/j + 1/k == 3/10}, {j, k}, Integers] – Bob Hanlon Aug 15 '15 at 14:50
• Hi Bob; Okay +1, they both seem to suggest that no matter what N is the coefficient taps out at 4. – bobbym Aug 15 '15 at 14:54