Jelly, 22 bytes

’1l·_+1l·’}\ð‘ị3ðṛ?
J’Ç€

Try it online!

Horrible recursive definition, I'm sure there's a better approach. Full program.

Jelly, 22 bytes

’1l·_+1l·’}\ð‘ị3ðṛ?
J’Ç€

Try it online!

Horrible recursive definition, I'm sure there's a better approach. Full program.Here 's a modified version to run as a test suite.

How it works

We just implement

$$T(n,k) = \begin{cases} a_n & \text{if } k = 0 \\ T(n-1,n-k) + T(n, k-1) & \text{otherwise} \end{cases}$$

then calculate \$T(i,i)\$ for each index \$i\$ of \$a\$

The first line defines \$T(n,k)\$, and the second calculates \$T(i,i)\$ for each index \$i\$.

’1l·_+1l·’}\ð‘ị3ðṛ? - Helper link. T(n,k).
 ṛ? - If k:
 ð - Then:
’ - n-1
 _ - n-k
 1l· - T(n-1, n-k)
 \ - Last two links as a dyad g(n,k):
 ’} - k-1
 1l· - T(n, k-1)
 + - T(n-1, n-k) + T(n, k-1)
 ð - Else:
 ‘ - n+1 (due to Jelly's 1 indexing)
 ị3 - Index into a
J’Ç€ - Main link. Takes a on the left
J - Indices of a
 ’ - Decrement to 0 index
 € - Over each index i:
 Ç - Yield T(i,i)