Bouncing Factorial

The Bouncing Factorial is a type of factorial that multiplies together the integers from 1 to some number n, then back to 1, then back to (n-1), then to 1, then to (n-2), and so on. It is to be denoted $$n\Lambda$$.

Example
So for instance, if you have the number 9, the bouncing factorial of 9 will be 1*2*3*4*5*6*7*8*9*8*7*6*5*4*3*2*1*2*3*4*5*6*7*8*7*6*5*4*3*2*1*2*3*4*5*6*7*6*5*4*3*2*1*2*3*4*5*6*5*4*3*2*1*2*3*4*5*4*3*2*1*2*3*4*3*2*1*2*3*2*1*2*1, or 9,278,496,603,801,318,870,491,332,608,000,000,000. Pictured on the left is a visualization of the bouncing factorial of 9, every new multiplication peak has been colored for clarity. The numbers above the peaks refer to their height in units. The x-axis may be thought of as time, and the y-axis as quantity, this graph illustrates how to calculate the bouncing factorial of 9, or the product of the quantities of all those little colored squares.

Formal Descriptions
The Bouncing Factorial of a number $$n$$can be formally defined as $$n(\prod_{i=1}^{n-1} i^{2n-2i+1})$$. This formula holds true for all values of $$n$$ greater than 1. When $$n$$ equals one, the bouncing factorial is 1.

It may also be recursively defined as $$Z_{n+1}={{(n+1)!^2}/n}*Z_n$$ where $$Z_1=1$$.

Primes
$$n\Lambda -1$$ is prime when $$n=1,2,4,...$$. As of the time this was written, no primes have been found of the form $$n\Lambda -1$$ for values of n less than or equal to 10.

Source
[1 ] Where a description of the function is to be found.