The Leibniz harmonic triangle
(1 + x)−2 = 1 – 2x + 3x^2 – 4x^3 + 5x^4 – 6x^5 + 7x^6 – 8x^7 +...The Leibniz harmonic triangle
The German polymath Gottfried Leibniz discovered a remarkable set of
numbers in the form of a triangle. The Leibniz numbers have a symmetry
relation about the vertical line. But unlike Pascal’s triangle, the number in one
row is obtained by adding the two numbers below it. For example 1/30 + 1/20 =
1/12. To construct this triangle we can progress from the top and move from left
to right by subtraction: we know 1/12 and 1/30 and so 1/12 − 1/30 = 1/20, the
number next to 1/30. You might have spotted that the outside diagonal is the
famous harmonic series
but the second diagonal is what is known as the Leibnizian serieswhich by some clever manipulation turns out to equal n/(n + 1). Just as we
did before, we can write these Leibnizian numbers as B(n,r) to stand for the nth
number in the rth row. They are related to the ordinary Pascal numbers C(n,r) by
the formula: