How to Calculate Leibniz Harmonic Triangle

Definition

The Leibniz harmonic triangle is a triangular piece of portions in which the furthest diagonals comprise of the reciprocals of the line numbers and every inward cell is indisputably the quality of the unit above minus the unit to the left.

Example:

Generate the first 3 rows of the Leibniz Harmonic Triangle (binomial coefficients)

Solution:

a_n,k = 1 / k (_nC_k) nC_k = n!/(n-k)! k!

Step 1:

a_(1,1) = 1 / 1*(₁C₁) (₁C₁) = 1!/(1-1)! 1! = 1 / 0! = 1 a_(1,1) = 1 / 1 * 1 = 1/1

Step 2:

a_(2,1) = 1 / 1 * (₂C₁) (₂C₁) = 2!/(2-1)! 1! = 2 / 1! = 2 a_(2,1) = 1 / 1 * 2 = 1/2 a_(2,2) = 1 / 2 * 1 = 1/2

Step 3:

The same steps follow upto n=3

Result:

1/1
1/2	1/2
1/3	1/6	1/3