Step by Step solution about How to resolve the algorithm Catalan numbers/Pascal's triangle step by step in the Java programming language

This Java program is designed to generate and print a sequence of numbers based on a specific pattern. Let's break down each part of the code:

1. Variables:

   • int N = 15;: This variable represents the desired length of the sequence to be generated. In this case, it's 15.
   • int[] t = new int[N + 2];: This creates an array t of size N + 2 to store the sequence of numbers. The extra two elements are added for convenience in the algorithm.

2. Initialization:

   • t[1] = 1;: The first element of the array t is initialized to 1. This serves as the starting point for the sequence.

3. Nested Loops: The code contains two nested loops responsible for generating the sequence:

   • Outer Loop (i loop):

     • This loop iterates from i = 1 to N (inclusive).

   • Inner Loop 1 (j loop - First Iteration):

     • In the first iteration of the inner loop, it increments each element t[j] from j = i down to j = 2 by adding the value of the previous element t[j - 1].

   • Assignment:

     • After the first inner loop, t[i + 1] is set to t[i].

   • Inner Loop 2 (j loop - Second Iteration):

     • In the second iteration of the inner loop, it increments each element t[j] from j = i + 1 down to j = 2 by adding the value of the previous element t[j - 1].

   • Printing:

     • Within the outer loop, it calculates and prints the difference between t[i + 1] and t[i], effectively printing the next number in the sequence.

4. Output:

   • The program generates and prints a sequence of numbers following the pattern defined by the nested loops.

In summary, this program initializes an array with a starting value, then iteratively calculates and stores values in the array based on the nesting loop structure. The final result is a sequence of numbers printed one after another, separated by spaces.

public class Test {
    public static void main(String[] args) {
        int N = 15;
        int[] t = new int[N + 2];
        t[1] = 1;

        for (int i = 1; i <= N; i++) {

            for (int j = i; j > 1; j--)
                t[j] = t[j] + t[j - 1];

            t[i + 1] = t[i];

            for (int j = i + 1; j > 1; j--)
                t[j] = t[j] + t[j - 1];

            System.out.printf("%d ", t[i + 1] - t[i]);
        }
    }
}