Consider the recurrence relation $$a_n=1 + \sum_{i=1}^{n-1}ia_{n-i}$$ with initial term $a_1=1$. What is $a_n$?
I tried to guess some closed formula from the first 6 terms, which are $1$, $2$, $5$, $13$, $24$, $79$, but no luck.
I tried to use generating functions to solve this sum and inner sum $\sum\limits_{i=1}^{n-1}ia_{n-i}$ seemed to be some kind of convolution of $a_n$ and $b_n=\langle1,1,\dots\rangle$, so this would give us $\sum\limits_{i=1}^{n-1}ia_{n-i}b_i$, but I can't solve this.
Lastly, I observed that $a_n - a_{n-1} = \sum\limits_{i=1}^{n-1}a_i$. But how am i supposed to use it?
I got all this information, but have no idea how to use it.