By writing [mathematical expression not reproducible] using a well-known identity of binomial coefficients
and then applying (4), we obtain
Beginning with the negative binomial coefficients
, we again find that greater cultural distance and existing immigrant stocks have negative and positive effects, respectively, on the level of the predicted immigrant stock.
1) is replaced here by the central binomial coefficient
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Equations (2)-(8) involve binomial coefficients
, whose values are large integer numbers for high upper indices, whereas powers of Courant numbers may be very small real numbers.
In this section we provide a formal definition of the two circulant determinant sequences with binomial coefficients
and derive the formula for their respective n-th term.
n] (I) is a partial sum of signed binomial coefficients
from the n-th row of the Pascal triangle (starting to count from n = 0).
one immediately sees that each binomial coefficient
of the form ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) is even, and if a and are even, then
For the sake of completeness we give, without proof, two particular theorems in regard to single binomial coefficients
and their integral representations.
where the coefficients 1, 2, 1 are binomial coefficients
Table 1 Regression and Negative Binomial Coefficients
for Bill Introduction in Four Regulatory Agencies, 1949-96 Bills (FGLS) Bills (Negbin) Independent Variables ([dagger]) ([double dagger]) Salience in noncomplex agencies [t.
In order to gain a sense of the magnitude of these effects, the negative binomial coefficients
can be transformed into incidence rate ratios.
Szalay studied the balancing binomial coefficients
, namely, the Diophantine equation