Intuitive Bernoulli numbers 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.
 A: 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):
$$
\sum_{0 \le k \le n - 1} k^m 
   = \frac{1}{m + 1} \sum_{0 \le k \le m} \binom{m + 1}{k} B_{m - k} n^k
$$
A: 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
