The Number of [Old-Time] Basketball games with Final Score n:n where the Home Team was never losing but also never ahead by more than w Points

The Number of [Old-Time] Basketball games with Final Score n:n where the Home Team was never losing but also never ahead by more than w Points
By Arvind AYYER and Doron ZEILBERGER
(Appeared in Electronic J. of Combinatorics 14(1) (2007), R19 [8 pp])
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Written: Oct. 24, 2006.
Life is complicated (non-linear) but sometimes it may be simplified (expressed in terms of linearity). It is a rather simple fact that Soccer is simple (in the sense of being "linearizable"), but it is extremely surprising [at least to us!] that basketball is "simple". Proving its simplicity, however, was not so simple. As Tim Gowers once said in a different context, "it is trivial, but not trivially trivial".
This paper gives birth to a new sequence of polynomials, AZ_w(z). AZ_w(z) is to Basketball what Tchebychev is to Soccer. Enjoy!

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