Closed form for this 2 variable recurrence?

Question

I'm trying to find a closed form for this two variable recurrence, but Wolfram Alpha does not seem to understand the input.

$$ \begin{cases} a_{0,1} = 1 \\ a_{0,i} = 0 \quad \forall i\neq1 \\ a_{j,k} = \sum_{i=1}^{k-1} \binom{i}{k-i} \cdot a_{(j-1),i} \end{cases} $$

with the convention that $$\binom{x}{y}=0 \quad \forall y \gt x $$

The problem arises from a combinatorial sequence; I'm trying to get the closed form to calculate the sequence terms quickly.

Answer 1

The values for $a_{j,j}$ are $0$. Values for $a_{j, j+1}$ are factorials $j!$, but I don't have a good idea for the others.

The table was generated by a home-made C# program making use of the System.Numerics.BigInteger library in Microsoft .Net