Lacsap's triangle

The following rules are used to construct Lacsap's triangle:

  1. Lacsap's triangle is a triangular array of numbers.
  2. There are \(n\) entries in the \(n\)th row. The entries are staggered relative to the previous row, similar to Pascal's triangle.
  3. The first and last entries in each row are 2.
  4. Each entry is equal to the product of the numbers above it to the left and to the right. If no entry is written, treat it as 1 (the multiplicative identity).

How many times does the value \(32768\) appear in Lacsap's triangle?

Details and assumptions

Using the rules, the first few rows are

\[ \begin{array} { c c c c c c c c c c c c c c c c c c c} & & & & & 2 & & & & & \\ & & & & 2 & & 2 & & & & & \\ & & & 2 & & 4 & & 2 & & & & \\ & & 2 & & 8 & & 8 & & 2 & & \\ & & & & & \vdots & & & & & \\ \end{array} \]

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