Related: Boustrophedonise, Output the Euler Numbers (Maybe a new golfing opportunity?)
Background
Boustrophedon transform (OEIS Wiki) is a kind of transformation on integer sequences. Given a sequence \$a_n\$, a triangular grid of numbers \$T_{n,k}\$ is formed through the following procedure, generating each row of numbers in the back-and-forth manner:
$$ \swarrow \color{red}{T_{0,0}} = a_0\\ \color{red}{T_{1,0}} = a_1 \rightarrow \color{red}{T_{1,1}} = T_{1,0}+T_{0,0} \searrow \\ \swarrow \color{red}{T_{2,2}} = T_{1,0}+T_{2,1} \leftarrow \color{red}{T_{2,1}} = T_{1,1}+T_{2,0} \leftarrow \color{red}{T_{2,0}} = a_2 \\ \color{red}{T_{3,0}} = a_3 \rightarrow \color{red}{T_{3,1}} = T_{3,0} + T_{2,2} \rightarrow \color{red}{T_{3,2}} = T_{3,1} + T_{2,1} \rightarrow \color{red}{T_{3,3}} = T_{3,2} + T_{2,0} \\ \cdots $$
In short, \$T_{n,k}\$ is defined via the following recurrence relation:
$$ \begin{align} T_{n,0} &= a_n \\ T_{n,k} &= T_{n,k-1} + T_{n-1,n-k} \quad \text{if} \; 0<k\le n \end{align} $$
Then the Boustrophedon transform \$b_n\$ of the input sequence \$a_n\$ is defined as \$b_n = T_{n,n}\$.
More information (explicit formula of coefficients and a PARI/gp program) can be found in the OEIS Wiki page linked above.
Task
Given a finite integer sequence, compute its Boustrophedon transform.
Standard code-golf rules apply. The shortest code in bytes wins.
Test cases
[10] -> [10] [0, 1, 2, 3, 4] -> [0, 1, 4, 12, 36] [0, 1, -1, 2, -3, 5, -8] -> [0, 1, 1, 2, 7, 15, 78] [1, -1, 1, -1, 1, -1, 1, -1] -> [1, 0, 0, 1, 0, 5, 10, 61] Brownie points for beating or matching my 10 bytes in ngn/k or 7 bytes in Jelly.
