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I have the following recursive formula: \begin{align} F(m,n) & = F(m,n - 1) + F(m - 1,n) - F(m - 1,n - 1 - m), \\ F(m,0) & = F \! \left( m,\frac{m (m + 1)}{2} \right) = 1, \\ F(m,i) & = 0 ~ \text{if} ~ i < 0 ~ \text{or} ~ i > \frac{m (m + 1)}{2}. \end{align} Is there a way to solve this recursive formula to obtain $ F(m,n) $ in closed form?

I tried the $ Z $-transform, but I got nothing.

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    $\begingroup$ Where did this come from? Context is always helpful. $\endgroup$
    – David Roberts ♦
    Commented Apr 9, 2015 at 0:46

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These are called Mahonian numbers, according to OEIS. https://oeis.org/A008302

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I am not sure if the formula suits your needs, but you may find it at: http://oeis.org/A181609. It represents some of the numbers in terms of Mahonian distribution.

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