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

Question

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?