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Leibniz Harmonic Triangle

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The Leibniz harmonic triangle is the number triangle given by

1/1 1/2 1/2 1/3 1/6 1/3 1/4 1/(12) 1/(12) 1/4 1/5 1/(20) 1/(30) 1/(20) 1/5
(1)

(OEIS A003506), where each fraction is the sum of numbers below it and the initial and final entries in the nth row are given by 1/n.

The terms are given by the recurrences

a_(n,1)=1/n
(2)
a_(n,k)=a_(n-1,k-1)-a_(n,k-1)
(3)

and explicitly by

a_(n,k)=1/(k(n; k))
(4)
=1/(n(n-1; k-1)),
(5)

where (n; k) is a binomial coefficient.

The denominators in the second diagonals are the pronic numbers 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, ... (OEIS A002378). A sorted list of all possible denominators in the triangle is given by 6, 12, 20, 30, 42, 56, 60, 72, 90, 105, 110, ... (OEIS A007622).

The row sums are given by 1, 1, 5/6, 2/3, 8/15, 13/30, 151/420, ... (OEIS A046878 and A046879). The sums of the denominators in the nth row are given by n·2^(n-1), giving the first few as 1, 4, 12, 32, 80, 192, 448, ... (OEIS A001787).

See also

Catalan's Triangle, Clark's Triangle, Euler's Number Triangle, Losanitsch's Triangle, Number Triangle, Pascal's Triangle, Seidel-Entringer-Arnold Triangle

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References

Sloane, N. J. A. Sequences A001787/M3444, A002378/M1581, A003506, A007622/M4096, A046878, and A046879 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Leibniz Harmonic Triangle

Cite this as:

Weisstein, Eric W. "Leibniz Harmonic Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LeibnizHarmonicTriangle.html

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