Presence of the monomial (−1)^{n−k} S2(n,k) x1^n in B_{n,k}(ψ1, …, ψ_{n−k+1})
Show that for all integers n ≥ 2 and 1 ≤ k < n, the Bell-polynomial-derived expression B_{n,k}(ψ1, …, ψ_{n−k+1}) always contains the monomial (−1)^{n−k} S2(n,k) x1^n, where S2(n,k) denotes the Stirling numbers of the second kind.
References
For the case $B_{n,k}$, one may conjecture in view of Table~\ref{tbl:Bnk} and further computed terms that $B_{n,k}$ always contains the term $(-1){n-k} \, S_2(n,k) \, x_1n$.
— Wilson's theorem modulo higher prime powers I: Fermat and Wilson quotients
(2509.05235 - Kellner, 5 Sep 2025) in Section 4, Remark