Focusing first on the results presented in column (a), we see that the estimated negative binomial coefficients
for the terms which interact the cultural distance variable and the dummy variables that identify the immigrants' skill levels are all negative and significantly different from zero.
Hence, by Lemma 3.3, we know that the total number of these structures is the central binomial coefficient
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
The invariant [N.sub.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).
The binomial coefficients
of words are closely related to the shuffle and infiltration products.
Just as with ordinary binomial coefficients
, q-binomial coefficients have multiple combinatorial interpretations and satisfy many identities, a few of which are highlighted below.
denote the usual q-analogs of numbers, factorials and binomial coefficients
An application of trinomial revision to the binomial coefficient
ratios in Theorem 3.2 is necessary in order to allow Maple to do the calculation.
Two new special power series, including the numbers of Lyndon words and binomial coefficients
, are defined in Section 3.
Diaz-Barrero  gave the following results, providing circular regions containing all the zeros of a polynomial in terms of the binomial coefficients
and Fibonacci's numbers.
We use the generalized binomial coefficients
[mathematical expression not reproducible] given by [mathematical expression not reproducible] for n [member of] Z with the convention that [mathematical expression not reproducible] for all integers i < 0.
Thus, loss factors L can be written in form of binomial coefficients
[L.sub.n,k]([eta]) = [[eta].sup.n][(1 -[eta]).sup.k-n]k!/((k -n)!n!), where [eta] is an efficiency coefficient that includes propagation losses and detection efficiency (beam splitter transmittance), k is the number of impinging photons, and n is the number of photons at the output of this beam splitter.
A note on balancing binomial coefficients
Shane CHERN Communicated by Masaki KASHIWARA, M.J.A.