# Definition:Leibniz Harmonic Triangle

## Definition

The Leibniz Harmonic Triangle is a triangular array where:

the zeroth element in the $n$th row, counting from $0$, is $\dfrac 1 {n + 1}$
subsequent elements in the $n$th row are the zeroth element divided by the corresponding element of Pascal's triangle:

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

### Row

Each of the horizontal lines of numbers corresponding to a given $n$ is known as the $n$th row of Leibniz harmonic triangle.

Hence the top row, containing a single $1$, is identified as the zeroth row, or row $0$.

### Column

Each of the vertical lines of numbers is known as the $m$th column of Leibniz harmonic triangle.

The leftmost column, containing the reciprocals of the non-negative integers, is identified as the zeroth column, or column $0$.

### Diagonal

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 $0$.

## Source of Name

This entry was named for Gottfried Wilhelm von Leibniz.