Sedrakyan's Power Sums Triangle

Self-Construction Principle (see the figure below: Sedrakyan's Power Sums Triangle).

The construction of the triangle is governed by the following rules:

• Initial Condition: The first row starts with $\dfrac{1}{2}$ and $\dfrac{1}{2}$, representing the formula for the sum of the first $n$ integers:
  $$1 + 2 + \dots + n = \frac{n(n+1)}{2} = \dfrac{\mathbf{1}}{\mathbf{2}} \cdot n + \dfrac{\mathbf{1}}{\mathbf{2}} \cdot n^{2}$$
• Rightmost Leading Diagonal: The coefficients along this diagonal are equal $\dfrac{1}{k+1}$.
• First Rightmost Dashed Diagonal: The coefficients along this diagonal remain constant at $\dfrac{1}{2}$.
• Row Initialization: Subsequent rows start with $B_{k}$, where $B_{k}$ are the Bernoulli numbers.
• Recursive Step: New row terms are built recursively by multiplying the previous diagonal term by $\dfrac{k}{p}$, where $k$ is the row index and $p$ is the power of $n$.