where
is a Roman factorial. The above expression is
read "Roman
choose ."
Whenever the binomial coefficient is defined
(i.e.,
or ),
the Roman coefficient agrees with it. However, the Roman coefficients are defined
for values for which the binomial coefficients
are not, e.g.,

(2)

(3)

where

(4)

The Roman coefficients also satisfy properties like those of the binomial
coefficient,

Loeb, D. E. "A Generalization of the Binomial Coefficients." 9 Feb 1995. http://arxiv.org/abs/math/9502218.Roman,
S. "The Logarithmic Binomial Formula." Amer. Math. Monthly99,
641-648, 1992.