Intuitive Bernoulli numbers

Question

Can somebody explain me or give me a link with a intuitive point of view of Bernoulli numbers?

I mean, somebody just saw a typical sequence of numbers that appears in some Taylor expansions, and them called them "Bernoulli numbers"?

How do they become with a method for finding these numbers? What's the intuition behind this?

I'm asking it because I can only find non-intuitive PDFs that only accept strange formulas and don't even explain them. And I wanted to know how to calculate a Bernoulli number.

Answer 1

Perhaps their homepage helps here...

One of the nice formulas involving them is the one for sum of powers discovered by Bernoulli (and from there they take their name): $$ \sum_{0 \le k \le n - 1} k^m = \frac{1}{m + 1} \sum_{0 \le k \le m} \binom{m + 1}{k} B_{m - k} n^k $$

Answer 2

The Wildberger/Kelly method for deriving Power Summation Formulas (Faulhaber Polynomials) including Bernoulli numbers.

I discovered the following formula that connects every term to the term immediately above.

                X   *   (Summation Index + 1)
                    Power of the Term + 1

NOTE : I CAN'T SHOW SUPERSCRIPT SO THE NUMBERS AFTER THE N'S ARE THE POWERS. THE LAST TERM IN EACH SUMMATION FORMULA IS THE Bernoulli number.

Deriving S2 from S1

S1  =   1/2n2  + 1/2n
        
S2  =   1/3n3  + 1/2n2  +  1/6n
        

Example S1 First Term (1/2 * (1 + 1)) / 2+1 = 1/3

Example S1 Second Term (1/2 * (1 + 1)) / 1+1 = 1/2

NB : All Summation formulas add up horizontally to 1, therefore we derive the 1/6th by subtracting the other terms from 1. Summations from S4 alternate minus and positive Bernoulli numbers.

1   -   (1/3 + 1/2)     =    1/6

Deriving S3 from S2

S2  =   1/3n3  + 1/2n2  + 1/6n

S3  =   1/4n4  + 1/2n3  + 1/4n2  -  0n

Example S2 First Term (1/3 * (2 + 1)) / 3+1 = 1/4

Example S2 Second Term (1/2 * (2 + 1)) / 2+1 = 1/2

Example S2 Third Term (1/6 * (2 + 1)) / 1+1 = 1/4

Deriving S4 from S3

S3  =   1/4n4  + 1/2n3  + 1/4n2  -  0n
            

S4  =   1/5n5  + 1/2n4  + 1/3n3   -   0n2  -  1/30n

Example S3 First Term (1/4 * (3 + 1)) / 4+1 = 1/5

Example S3 Second Term (1/2 * (3 + 1) / 3+1 = 1/2

Example S3 Third Term (1/4 * (3 + 1)) / 2+1 = 1/3

More details here,

https://books.apple.com/us/book/wildberger-kelly-method-for-calculating-faulhaber-polynomials/id1567798642?ls=1