C_{0} = C_{1} = 1 and C_{n} = \frac{(2 n)!}{((n + 1)! n!)} = \sum_{i = 0}^{n - 1} C_{i} C_{n - 1 - i} = \frac{2 (2 n - 1)}{n + 1} C_{n - 1} for n > 2

Hint

Start with the foundation – the base cases, which are the simplest solutions. Then, layer by layer, use the results of the previous layers to calculate the solutions for the next level. This way, we systematically build up the entire solution, avoiding redundant calculations and ensuring efficiency. It is like building a pyramid.

The underlying calculation logic remains fundamentally identical to the original recursive approach, albeit executed in reverse order.

# Python implementation

def C(n):
  cache = [0 for _ in range(n + 1)]

  cache[0] = cache[1] = 1

  for i in range(2, n + 1):
    result = 0

    for j in range(i):
      result += cache[j] * cache[i - 1 - j]
    
    cache[i] = result

  return cache[n]

n = 5

print(C(n))

// Javascript implementation

function C(n) {
  const cache = new Array(n + 1).fill(0);

  cache[0] = cache[1] = 1;

  for (let i = 2; i < n + 1; i++) {
    let result = 0;

    for (let j = 0; j < i; j++) {
      result += cache[j] * cache[i - 1 - j];
    }

    cache[i] = result;
  }

  return cache[n];
}

const n = 5;

console.log(C(n));

Find Nth Catalan Number (Tabulation)

Hint