i have a quite complicated formula in d dimensions and i am trying to find a recursion rule by $n$ for it.
$S^d_n = \displaystyle \sum_{j_1,\dotsc,j_{d-1} \in \mathbb{N}_0\\ \sum_{i=1}^{d-1}j_i \leq n} \sum_{i_1,\dotsc,i_{d-1} \in \{0,1\}} (-1)^{i_1+\dotsc,i_{d-1}} f\left(j_1-i_1,\dotsc,j_{d-1}-i_{d-1},n-\sum_{i=1}^{d-1}j_i\right)$
* Edit For $f$ it holds that $f(j_1,\dotsc,j_d) = 0$ if any $j_1,\dotsc,j_d = 0$ Edit *
For $d=2$ I found the following rule:
$S^2_n = \displaystyle \sum_{j=0}^n f(j,n-j) - S^2_{n-1-j}$
I would really appreciate your help.