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Leibniz Harmonic Triangle Calculator

This calculator generates values from the Leibniz Harmonic Triangle, a triangular array similar to Pascal’s Triangle, where each term is calculated using the formula $$aₙ,ₖ = 1 / k × C(n, k)$$. It is useful in combinatorics and series expansions, especially involving harmonic numbers and integrals.

Leibniz Harmonic Triangle Calculator

Input Fields
If enabled, the result will update automatically when you change any value.

Leibniz Harmonic Triangle Formula

Formula
$$a_{n,k} = \frac{1}{k} \cdot \binom{n}{k}$$

Where

  • $$a_{n,k}$$ is the value at row $$n$$, position $$k$$
  • $$\binom{n}{k}$$ is the binomial coefficient “n choose k”
  • $$k$$ must be greater than 0

The Leibniz Harmonic Triangle is a triangular array of rational numbers. Each entry is derived from a modified binomial coefficient: instead of simply using $$\binom{n}{k}$$, the value is scaled by $$\frac{1}{k}$$. This structure appears in the study of harmonic series, definite integrals, and generating functions. Unlike Pascal’s triangle, this array deals with weighted combinatorics and highlights the relationship between binomial coefficients and harmonic behavior. You can compute a specific term by inputting the row number n and position $$k$$ (1-based index).

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