Bernoulli Formulas

I want the sum 5 natural numbers squared. ( In effect,
I am looking for 30).

The long way:
                          0^2 + 1^2 +2^2 + 3^2 + 4^2
                       = 0 + 1 + 4 + 9 + 16
                       = 30

Using Bernoulli formulas, I have three terms to add:

1/3 (n^3)  - 1/2(n^2) + 1/6(n)
= 1/3(5^3) - 1/2(5^2) + 1/6(5)
= 125/3 -25/2 +5/6
= 250/6 - 75/6 + 5/6
= 180/6
= 30

Below, the well known Bernoulli numbers. They are, in effect, the last

coefficient from the lines in the formula table. The odd-numbered lines three and up are

listed as 0, because that is the value they evaluate to.

source: Wikipedia

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To appreciate the computational models, one needs to be familiar with Pascal's

triangle.

 The first number in the parenthesis is the line; the bottom one is the column!

Should I wish to calculate (a + b)^5 with it, I would go to line 6. The coefficients

go from left to right, with powers going down on a, and up on b.

https://omnilogie.fr/O/Le_triangle_de_Pascal,_outil_bien_pratique_en_Maths_!

Mathematically, one can find the coefficients within Pascal's triangle

by the following. These are the binomial coefficients:

source: Wikipedia

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