Category:Element of Leibniz Harmonic Triangle as Sum of Elements on Diagonal from Below

This category contains pages concerning Element of Leibniz Harmonic Triangle as Sum of Elements on Diagonal from Below:

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}$

Let $\tuple {n, m}$ be the element in the $n$th row and $m$th column.

Then:

$\tuple {n, m} = \ds \sum_{k \mathop \ge 0} \tuple {n + 1 + k, m + k}$