The author considers Latin rectangles of size \(3\times k\). A Latin rectangle is called even (resp. odd) if the product of permutations of the rows is even (resp. odd). The generating function for the numbers \(S_{3,k}\) of differences between even and odd Latin rectangles of size \(3\times k\) is known to be \(\sum_kS_{3,k} (-1)^{n-1} {2t^n\over (n-2)n!} = e^{2((\text{acrtanh} t)-t)}\), see L. Habsieger, J. M. C. Janssen, Ann. Sci. Math. Qué. 19, No. 1, 69-77 (1995; Zbl 0832.05014). It is proven that this formula is equivalent to \(1+ \sum S_{3,k} {t^n \over n!} = e^{2t[{(1-t)^2 \over 1+t} + {t\over (1+t)^2}]}\), which can be obtained as a special case of formulas of G. E. Andrews, I. P. Goulden and D. M. Jackson [Trans. Am. Math. Soc. 310, No. 2, 805-820 (1988; Zbl 0707.05061)].