MATHEMATICS BINOMIAL FORMAULA

//******************************************************************************
// MathsPhysic Code
// Binomial formula
//******************************************************************************

//today is maths and we will try to copute (a + b)^n  = binomial theorem (Newton)
#include <stdio.h>
//we will use recursion to calculta the Pascal Triangle coefficient
int pascal(int n, int p){
    if(p == 0 || n == p)
        return 1;
    return pascal( n -1, p - 1 )  + pascal(n - 1,  p  ); //we add triangle previous line coef (see pascal)
}
//n - 1 is previous line and p-1 the previous column of current n box
// p is current box column

//now, let's compute Newton binomial using Pascal coefficient
int  main(int argc, char **argv){ // i will use command line
    if (argc == 1)
        return (-1);          //Error
                              //Usage example : ./a.out (a + b)^3
    int n = argv[1][6] - '0'; //argv[1][6] equal 3 in this example (above), 7th character
                              // argv[1][6] - '0' , i use ascii table to convert it to integer
                              // '0' is 48 in ascii table so it convert to int
    int k = n;

    printf("%s = " , argv[1]);
    int operator = argv[1][2] != '-'; //check if it is (a-b)^n or (a + b)^n
    for (int i = 0; i<= n; i++){//i will print in color , to save line , i will use ternary operator
        printf("\x1B[31m%d", pascal( n , i )); //pascal coefficient to print
        printf("\x1B[32ma^%d", k--); //decrement l in each step for new recursion
        printf("\x1B[33mb^%d", i); //printf coefficient in color in natural way

        if(i < n) // i use ternary operator to save line : it only to check the sign +/-
            operator || i&1 ? printf(" + ") : printf(" -");
    }

}