Log Concavity Involving the Number of Paths from the Origin to Points Along the Line \((a, b,c, d)+t(1,-1,1,-1)\)

This paper examines the numbers of lattice paths of length \(n\) from the origin to integer points along the line \((a,b,c,d) + t(1,-1,1,-1)\). These numbers form a sequence which this paper shows is log concave, and for sufficiently large values of \(n\), the location of the maximum of this sequence is shown. This paper also shows unimodality of such sequences for other lines provided that \(n\) is sufficiently large.