So here we are needing to find out how to derive the z numbers so that we can write out the summation formula for any arbitrary power. My first method for finding the equations took a lot of work until I realized that the equations follow a regular pattern much to my relief.
Here are notes on my original method and then next I will detail my current method.
Original Method
To find the formula for the z (the nth column coefficient for the power summation formulas):
- Find the lcm of the denominators.
- Make a list of results from lcm divided by the denominators multiplied by the numerators.
- Make a list of those results divided by n (n being the nth power).
- Find formula for those numbers.
Example z4, lcm is 720
4 24 6
5 60 12
6 120 20
7 210 30
8 336 42
The formula for the third column is the summation of the evens with n being substituted with (n-2) to account for the alignment and then that formula is multiplied with the actual n from column one and then everything is divided by 720.
Current Method
Starting from z3 the formulas follow a regular pattern. The formula for the xth z number can be written out by using the formula:
So now we just need a way of deriving the constant for a given z formula. I have a method for doing this though it relies on knowing the summation formula for a power with enough terms to contain one in the column for the z constant we are looking for.
Finding the Constant
To demonstrate this method we will work through an example for finding the constant for the 10th z number.
We first write out the formula with a place holder value of for C.
Next we need the formula for a power summation with at least 10 terms, the first of which is:
With a C of
So to get the correct value of C we take the 10th coefficient from our summation formula and divide that by the number we just got.
The final formula:
So now we must ask the question what is the formula for the C values of the z functions? Well I had to know so I googled and there was something about Bernoulli numbers and a whole bunch of mathematical notation that is meaningless to me. Though that is good news because I can go about finding a formula for the C values much as I have with the rest of my math, a fun riddle to be solved.