So for the signed stirling numbers of the first kind, I wanted a formula for the diagonals
On Wikipedia they describe the formula as the below. I don't quite understand how the indexing works for the sum (why is there a sum inside the index and what is the small k's)
Here the link to it. Any help is much appreciated!
The notation
$$\sum_{0\le k_1,\ldots,k_p:\sum_1^pmk_m=p}f(k_1,\ldots,k_p)$$
means that we’re summing the general term $f(k_1,\ldots,k_p)$ over all $p$-tuples $\langle k_1,\ldots,k_p\rangle$ of non-negative integers such that $\sum_{m=1}^pmk_m=p$. Each of these $p$-tuples describes a partition of $p$: specifically, each $k_m$ for $m=1,\ldots,p$ is the number of parts of size $m$ in the partition. Thus, if $p=11$, the $11$-tuple $\langle 3,2,0,1,0,0,0,0,0,0,0\rangle$ describes the partition $11=4+2+2+1+1+1$.
The summation could also be written
$$\sum\left\{f(k_1,\ldots,k_p):0\le k_1,\ldots,k_p\text{ and }\sum_{m=1}^pmk_m=p\right\}\,.$$