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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