  For any integer $k\geq2$, we prove combinatorially the following Euler
(binomial) transformation identity $$ \NC_{n+1}^{(k)}(t)=t\sum_{i=0}^n{n\choose
i}\NW_{i}^{(k)}(t), $$ where $\NC_{m}^{(k)}(t)$ (resp.~$\NW_{m}^{(k)}(t)$) is
the sum of weights, $t^\text{number of blocks}$, of partitions of
$\{1,\ldots,m\}$ without $k$-crossings (resp.~enhanced $k$-crossings). The
special $k=2$ and $t=1$ case, asserting the Euler transformation of Motzkin
numbers are Catalan numbers, was discovered by Donaghey 1977. The result for
$k=3$ and $t=1$, arising naturally in a recent study of pattern avoidance in
ascent sequences and inversion sequences, was proved only analytically.
