Is it a Pascal Prime? It is well known that odd primes will appear in Pascal's triangle exactly twice. However, not all numbers that appear exactly twice in Pascal's triangle are prime. We'll call these numbers Pascal primes. 

Pascal primes are composite numbers that appear exactly twice in Pascal's triangle. The first few Pascal primes are

4, 8, 9, 12, 14, 16, 18, ...


Your challenge is to take a positive integer n as input and output true or false, depending on if n is a Pascal prime or not. This is code-golf, so the shortest program wins!
Wolfram Language (Mathematica), 45 bytes

CompositeQ@#&amp;&amp;Binomial~Array~{#-1,#}~FreeQ~#&amp;


Try it online!

Every composite number n appears exactly twice on row n and cannot appear afterwards. So the condition for Pascal primes is that they don't appear at all within the first n-1 rows.

As far as I can tell, this is shorter than checking that it appears exactly twice within the first n rows and being able to use !PrimeQ instead.