how Bernoulli are those Numbers, anyway?

I’m publishing this article even though I’m not at the point where I can fact check it as thoroughly as I’d like. I’m aiming to have a better understanding of the “Bernoulli Numbers“ but I’m not quite there yet. Right now I’m knee-deep in the Simplex Method, and I’m finding it utterly fascinating… like where-has-this-algorithm-been-all-my-life kind of fascinating. I’m gradually making my way towards the so-called “Bernoulli Numbers.“ I want to remember that this is a goal of mine.

Though the sums of integer powers have been of interest to mathematicians since antiquity, it wasn’t until 1683 that Seki Takakazu, a Japanese mathematician, discovered the resultant in a in a series that would eventually be known as the ‘Bernoulli numbers.’ By 1710, Seki completed his work on the determinants of this series. Titled Katsuyō Sanpō, and published in 1712 (almost a year before the publication of Jakob Bernoulli’s formula in Ars Conjectandi), his work demonstrates the tabulation of binomial coefficients as well as Bernoulli numbers. Unfortunately, no formula based on this sequence of constants is presented in Katsuyō Sanpō.

In spite of his death in 1705, Bernoulli was able to independently realize the existence of the same sequence of constants and provide a uniform formula for all sums of powers. This formula is generally taken to be the most useful to date however no further computation of the sequence was completed until 1748. This is when Leonhard Euler succeeds in computing the 30th coefficient in the series. Once again, further computation of the series grinds to a halt. This time almost a full century passes, then along comes Lady Ada Lovelace’s Note G.

If it weren’t for a very particular addition to the translation of Luigi Menabrea’s “Sketch of The Analytical Engine”, who knows how long further computation of this series would’ve taken. Published in 1843, Ada Lovelace’s Note G outlines a detailed plan for using Charles Babbage’s Analytical Engine and punched cards to weave out the sequence of numbers. Not only is this considered the first computer program, it may also be the first time a recursive algorithm for calculating Bernoulli numbers is described.