My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too. Or just jump to my GitHub repository.

       
#include <iostream>
       
#include <vector>
       
#include <set>
       
 
       
// return true if x cannot be divided by p^2 for all primes 2 <= p <= maxSquare

       
bool isSquarefree(unsigned long long x, unsigned int maxSquare)
       
{
       
  // instead of a proper prime sieve, just perform trial division of all numbers
       
  for (unsigned int p = 2; p <= maxSquare; p++)
       
    if (x % (p*p) == 0)
       
      return false;
       
 
       
  // yes, squarefree
       
  return true;
       
}
       
 
       
int main()
       
{
       
  // size of the triangle
       
  unsigned int numRows = 51;
       
  std::cin >> numRows;
       
 
       
  // all squarefree numbers
       
  std::set<unsigned long long> squareFree = { 1 };
       
 
       
  // initial row
       
  std::vector<unsigned long long> current = { 1 };
       
  // ... and compute all further rows
       
  for (unsigned int row = 1; row < numRows; row++)
       
  {
       
    // last and first element is always 1
       
    std::vector<unsigned long long> next(current.size() + 1, 1);
       
 
       
    // fill remaining cells
       
    for (unsigned int column = 1; column < next.size() - 1; column++)
       
      next[column] = current[column - 1] + current[column];
       
 
       
    // symmetric: check only half of the triangle, skip borders, too (always 1)
       
    for (unsigned int i = 1; i <= next.size() / 2; i++)
       
    {
       
      auto x = next[i];
       
      if (isSquarefree(x, numRows))
       
        squareFree.insert(x); // std::set prevents duplicates
       
    }
       
 
       
    // next iteration
       
    current = std::move(next);
       
  }
       
 
       
  // find sum of all squarefree numbers
       
  unsigned long long sum = 0;
       
  for (auto x : squareFree)
       
    sum += x;
       
 
       
  // display result
       
  std::cout << sum << std::endl;
       
  return 0;
       
}

This solution contains 10 empty lines, 13 comments and 3 preprocessor commands.