6
votes
3answers
388 views

$n$-ary trees with $k$-internal nodes - Catalan numbers

It is known that the Catalan numbers count the number of binary trees with $k$-internal nodes. I was thinking of how to count ternary trees or in general $n$-ary trees with $k$ internal nodes and got ...
9
votes
2answers
645 views

“Ballot numbers” sum up to Catalan numbers

Summing certain numbers and comparing the results with OEIS, I found that $ \sum_{k=1}^n \frac{k^2}{n} \binom{2n-k-1}{n-1} = C_{n+1} - C_{n}, $ where $C_n$ denotes the $n^{\textrm{th}}$ Catalan ...