# Is there a deep connection between the urn problem and the sum of powers?

The urn problem goes as follows: How many ways are there to place $k$ indistinguishable balls into $n$ distinguishable urns? The solution to the problem being $\binom {n+k-1}{k}$ (or $\binom {n+k-1}{n-1}$).

Upon typing the solution into Wolfram Alpha I found the following table:

This table, I noticed, exhibits uncanny similarity to the table of the sums of powers $\sum _{i=1}^{n}i^{k}$:

My question is, is this similarity accidental, or is there a more deep connection between the two? And if there is such connection, is there an intuitive way of deriving the first from the second and vise versa?

• It breaks down completely for $k=4$ (upper table) so I'm thinking that it's just coincidence. Factors such as $1/2$ and $1/6$ are very common. Dec 24, 2016 at 15:34
• Well, for one thing, some properties of the binomial distribution are derived using the binomial theorem, so a connection shouldn't be too surprising. However, this answer doesn't really get at whether there is a deep connection. Dec 24, 2016 at 15:35
• see Bernoulli numbers. Dec 24, 2016 at 15:51