We introduce a family of rational polynomials, for integer $n \ge 0$, as
$$ r_{n}(x) = \sum_{k=0}^n \frac{\sum_{j=0}^k x^j \binom{k}{j}(j+1)^n}{k+1}. $$
Let $\operatorname{B}_n(x)$ denote the Bernoulli polynomials, let us agree to write $ \operatorname{B}_n $ for $ \operatorname{B}_n(1) $ and define the harmonic numbers $\operatorname{H}_n = \sum_{k=0}^n \frac{1}{k+1}$. We observe
$$ {r}_{n}(-1) = \operatorname{B}_n, $$ $$ {r}_{n}(0) = \operatorname{H}_n. $$
Because of this property, we give the polynomials the (tentative) name harmonic Bernoulli polynomials. The formula for $r_{n}(x)$ is a variant of a well-known representation of the Bernoulli numbers, for example, discussed by M. Riedel in this MSE question and recently used in a formal verification program.
Note that the integral of $r_n(x)$ over the left unit interval shifts the Bernoulli numbers one place to the right, $$ \int_{-1}^{0} r_{n+1}(x) = r_{n}(-1) \quad (n \ge 0). $$
The common denominator of the polynomials, that is the least common multiple of the denominators of the polynomial coefficients, is $\operatorname{lcm} \{k + 1 : 0 \le k \le n \} $, which we denote with LCM$(n)$.
$$ \operatorname{lcm}\{ \operatorname{denom}([x^k]\, r_n(x)) : 0 \le k \le n \} = \operatorname{LCM}(n) \quad (n \ge 0). $$
This leads to the following infinite lower triangular integer matrix $\operatorname{T}$ for $0 \le k \le n$: $$ \operatorname{T}(n, k) = \operatorname{LCM}(n) [x^k] r_{n}(x) $$
$$ \begin{matrix} 1 \\ 3 \quad 2 \\ 11 \quad 28 \quad 18 \\ 25 \quad 184 \quad 351 \quad 192 \\ 137 \quad 2608 \quad 11097 \quad 16128 \quad 7500 \\ 147 \quad 6816 \quad 57591 \quad 166912 \quad 193750 \quad 77760 \\ \ldots \end{matrix} $$
The triangle has the first column $\operatorname{T}(n,0) = \operatorname{LCM}(n) \operatorname{H}(n)$, and the main diagonal $\operatorname{T}(n,n) = \operatorname{LCM}(n)\, (n + 1)^{n - 1}$. But the key statement is
$$ \operatorname{B}_n = \frac{\sum_{k=0}^n (-1)^k\operatorname{T}(n,k)}{\operatorname{LCM}(n)}. $$
Such a representation of the Bernoulli numbers suggests asking for a combinatorial interpretation of the $\operatorname{T}(n, k)$.
Which combinatorial objects does $\operatorname{T}(n,k)$ count?
