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

- Lacsap's triangle is a triangular array of numbers.
- There are \(n\) entries in the \(n\)th row. The entries are staggered relative to the previous row, similar to Pascal's triangle.
- The first and last entries in each row are 2.
- 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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