Motivation: We informally call an infinite lower triangular matrix $\operatorname{T}(n, k)$ of integers a combinatorial triangle of a sequence of integers or rational numbers if it can be obtained from it by some simple operation (e.g., by summation or alternating summation of the rows), in the case of a rational sequence with an additional normalization.
For example, the triangle of the Stirling set numbers is a combinatorial triangle for the Bell numbers, Euler's triangle (Eulerian numbers) for the factorial numbers, or Pascal's triangle for the powers of 2. Of course, there can be a multitude of triangles that give a combinatorial interpretation of a sequence in this way. Are there such triangles for the Bernoulli numbers?
Construction: The first-order Eulerian numbers are defined as
\begin{equation} \left\langle n\atop k \right\rangle = (k+1) \left\langle n-1\atop k \right\rangle + (n-k) \left\langle n-1\atop k-1 \right\rangle, \end{equation}
with boundary conditions $\left\langle 0\atop 0 \right\rangle=1$, $\left\langle n\atop k \right\rangle =0$ for $k<0$ or $k > n$.
Further we set, for integer $0 \le k \le n$:
\begin{equation} \left[ n \atop k \right] = \frac{\operatorname{lcm}_{j=0}^n \binom{n}{j}}{\binom{n}{k}} \end{equation}
\begin{equation} \operatorname{T}_{n, k} = \left[ n \atop k \right] \left\langle n \atop k \right\rangle \end{equation}
The first few values of the triangle $\operatorname{T}(n,k)$ are:
\begin{equation} \begin{matrix} 1 \\ 1 \quad 0 \\ 2 \quad 1 \quad 0 \\ 3 \quad 4 \quad 1 \quad 0 \\ 12 \quad 33 \quad 22 \quad 3 \quad 0 \\ 10 \quad 52 \quad 66 \quad 26 \quad 2 \quad 0 \\ \ldots \end{matrix} \end{equation}
The Bernoulli numbers ($ \operatorname{B}_1 = 1/2 $) can be obtained for $n \ge 0$ with: \begin{equation} \operatorname{B}_n = \frac{\sum_{k=0}^n (-1)^k \operatorname{T}_{n,k}} { \operatorname{lcm}_{k=0}^n (k + 1)}. \end{equation} \begin{equation} \operatorname{B}_n = \frac11, \frac12, \frac16, 0, -\frac{2}{60}, 0, \frac{10}{420}, 0, -\frac{84}{2520}, 0, \frac{2100}{27720}, 0, -\frac{91212}{360360}, \ldots \end{equation}
Which combinatorial objects does $\operatorname{T}(n,k)$ count?
