# Match the Product

$\large P = \prod_{k=0}^{m-1}\left[m(m+1)-2T_k\right]$

where $$T_k$$ is the $$k$$th triangular number and $$m=\left \lceil \dfrac n2 \right \rceil$$

Then $$P$$ equals which of the following:

1. $$\displaystyle 2\prod_{x=0}^{m-1}(m^2-x^2)$$ for odd $$n$$
2. $$2n!$$
3. $$\displaystyle 2\prod_{x=0}^{m-1}(m^2-x^2)$$ for even $$n$$
4. $$n!$$

Notations:

• $$T_k=\dfrac {k(k+1)}2$$, where $$k$$ is a non-negative integer, denotes the triangular numbers 0,1,3,6,10,15....
• $$\lceil \cdot \rceil$$ denotes the ceiling function.
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