Previously, we have explored methods to compute the essence of power functions $Ȣx^n$ which involves solving a large system of linear equations. This method is equivalent to solving for the inverse of a large $n\times n$ matrix where entries are values of pascal's triangle. Though the matrix method allow us to solve for large number of essences at once, it does not extend easily to solve for next iterations of essence coefficients. Rather than reusing the values we have already solved for, we will have to solve for inverse of a separate larger matrix again. Here we will introduce an iterative method for solving for these coefficients. Chapter 0: Recap Let us first remind ourselves of the definition of essence. For a function $f(x)$, we want to find the transformation $Ȣf(x)$ such that we are able to 'smooth out' its series: $$\sum_{i=a}^b f(i) = \int_{a-1}^b Ȣf(x) dx$$ For example, we can solve for the following functions: $$\begin{align*}Ȣ1 &= 1 \\ Ȣx &= x +
Partitions are number of ways an integer can be represented as sum of positive integers. We can ask, for example, what is the partition of 5? If we write out every possible combination, $$\begin{align*} 5 &= 1+1+1+1+1 \\ &= 1+1+1+2\\ &= 1+1+3\\ &= 1+4\\ &= 1+2+2\\ &= 2+3\\ &= 5 \end{align*} $$ we can see that partition of 5 is 7. One will immediately notice, however, that this is not the most efficient approach to answering the question: not only does partition grow quickly, attempting to organize and not miss or repeat an algebraic expression becomes messy and impractical. Chapter 1: Partition Tree A cleaner, but still not the best, approach would be to use tree diagrams to represent the partitions of a number. In a Partition Tree of $n$, every path to a terminating node will add up to the number $n$. And also, every child of a node (every nodes below a given parent node) will always be greater than or equal to the parent node in order to not