  Nevanlinna-Herglotz functions play a fundamental role for the study of
infinitely divisible distributions in free probability. In the present paper we
study the role of the tangent function, which is a fundamental
Herglotz-Nevanlinna function and related functions in free probability. To be
specific, we show that the function $$ \frac{\tan z}{1-x\tan z} $$ of Carlitz
and Scoville describes the limit distribution of sums of free commutators and
anticommutators and thus the free cumulants are given by the Euler zigzag
numbers.
