Leibniz harmonic triangle

The harmonic triangle Leibniz or cal Harmonic Triangle of Gottfried Wilhelm Leibniz is analogous to Pascal 's triangle:

• The n-th row begins and ends at the edge of
• Each number is the sum of the two numbers under their

The entries are indicated by the symbol, wherein the numbering of the rows and columns with 1 starts (this is handled in the literature are not uniform (starting at 0 and at 1 ) ).

It is the recursion

A connection with the binomial coefficients of Pascal's triangle is given by

Because this results in the sum of the denominator in the n- th row. For example.

For the sum of a diagonal arises because

The telescopic sum

Because of the unit fractions followed by crossing the line of Leibniz:

History

Christiaan Huygens in 1672 his young friend Leibniz asked the summation of the reciprocal triangular numbers as a task:

It indicates the sum 2. During his stay in Paris, he worked extensively with the writings of Blaise Pascal. In a later version of his Historia et Origo he is Pascal's triangle compared with its harmonious triangle. The series is then obtained from the general series for n = 2