Definition:Leibniz Harmonic Triangle/Diagonal

Definition

Consider the Leibniz harmonic triangle:

$\begin{array}{r|rrrrrr}

n & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline 0 & \frac 1 1 \\ 1 & \frac 1 2 & \frac 1 2 \\ 2 & \frac 1 3 & \frac 1 6 & \frac 1 3 \\ 3 & \frac 1 4 & \frac 1 {12} & \frac 1 {12} & \frac 1 4 \\ 4 & \frac 1 5 & \frac 1 {20} & \frac 1 {30} & \frac 1 {20} & \frac 1 5 \\ 5 & \frac 1 6 & \frac 1 {30} & \frac 1 {60} & \frac 1 {60} & \frac 1 {30} & \frac 1 6 \\ \end{array}$

The $n$th diagonal of Leibniz harmonic triangle consists of the entries in row $n + m$ and column $m$ for $m \ge 0$:

$\left({n, 0}\right), \left({n + 1, 1}\right), \left({n + 2, 2}\right), \ldots$

Hence the diagonal leading down and to the right from $\left({0, 0}\right)$, containing the reciprocals of the non-negative integers, is identified as the zeroth diagonal, or diagonal $0$.

Also see

• Definition:Row of Leibniz Harmonic Triangle
• Definition:Column of Leibniz Harmonic Triangle

Sources

• Weisstein, Eric W. "Leibniz Harmonic Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LeibnizHarmonicTriangle.html