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

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?

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  • $\begingroup$ 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. $\endgroup$ Dec 24, 2016 at 15:34
  • $\begingroup$ 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. $\endgroup$ Dec 24, 2016 at 15:35
  • $\begingroup$ see Bernoulli numbers. $\endgroup$
    – Phicar
    Dec 24, 2016 at 15:51

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