How to Calculate Leibniz Harmonic Triangle

How to Calculate Leibniz Harmonic Triangle - Definition, Formula, Example

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.

Formula

an,k = 1 / k * (nCk) Where n - no of rows k - no of colums for each row C - binomial coefficient
Example:

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

Solution:

an,k = 1 / k (nCk) nCk = n!/(n-k)! k!

a(1,1)    
a(2,1) a(2,2)  
a(3,1) a(3,2) a(3,3)
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

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