Definition:Leibniz Harmonic Triangle
This page is about Leibniz Harmonic Triangle. For other uses, see Harmonic.
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 $0$.
Also see
Source of Name
This entry was named for Gottfried Wilhelm von Leibniz.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $35$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $35$
- Weisstein, Eric W. "Leibniz Harmonic Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LeibnizHarmonicTriangle.html