This algorithm helps to generate the sum equation of the power function:
usage: posum.py [-h] [-d DEGREE] [-v] [-s SHOW]
optional arguments:
-h, --help show this help message and exit
-d DEGREE, --degree DEGREE
-v, --version show version
-s SHOW, --show SHOW only showing the formula (f), triagle matrix (m) or both (b)[default]?
If we want to know the equation of the sum function of the nth power, we run the python script and enter the degree n of the sum function.
python posum.py -d <the degree n>
The result consists of the power sum formula of n degree and the triangle matrix. Besides, wanting to show either. Let's use the argument -s
Need help to use
python posum.py -h
To show version of algorithm.
python posum.py -v
For example, when we want to find the sum function of the power of 6, we run the command and enter the degree of the function as 6.
#input
python posum.py -d 6
#output
Sum formula of x^ 6 is: +1/42*x^1+0*x^2-1/6*x^3+0*x^4+1/2*x^5+1/2*x^6+1/7*x^7
Matrix triagle is:
1 2 3 4 5 6 7
0 1
1 1/2 1/2
2 1/6 1/2 1/3
3 0 1/4 1/2 1/4
4 -1/30 0 1/3 1/2 1/5
5 0 -1/12 0 5/12 1/2 1/6
6 1/42 0 -1/6 0 1/2 1/2 1/7
So the equation sum of the power of the sixth power has the form:
The algorithm works by forming a triangular matrix with each value of the matrix being a coefficient of the sum equation. The coefficients of the equation of degree n are the values of the n-th row of the matrix. Values of the matrix is calculated according to the system of equations:
$$\left{ \begin{array}{l} {a_{01}} = 1\ {a_{ij}} = \frac{i}{j} \times {a_{{(i - 1)(j - 1)}}} {\rm{ ,}}\forall i\ge 1,\forall j\ge 2\ \sum\limits{k = 1}^j {{a_{ik}} = 1{\rm{ ,}}\forall i,j} \end{array} \right.$$