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.

3, we know that the total number of these structures is the central

binomial coefficient [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

n] (I) is indeed an extension of the usual notion of the

binomial coefficient.

2 Classical

binomial coefficients of words, shuffle and infiltration

2 where we require a triple alternating sum of products of

binomial coefficients to get an expression for the number of words of [[2k].

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 respectively.

This paper makes use of the recently introduced technique of [gamma]-operators to evaluate the Hankel determinant with

binomial coefficient entries [a.

An application of trinomial revision to the

binomial coefficient ratios in Theorem 3.

The terms [alpha] and [beta] are typical expressions to which 'Rice's method' can be applied, due to the presence of the

binomial coefficient inside an alternating sum.

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.

Topics encompass counting and proofs, sets and logic, graphs and functions, induction, algorithms with ciphers,

binomial coefficients and Pascal's triangle, counting techniques, recurrences, counting and geometry, trees, Euler's formula and applications, graph traversals, graph coloring, probability and expectation, and cardinality.

A note on balancing

binomial coefficients Shane CHERN Communicated by Masaki KASHIWARA, M.

Although the new DGF expression has a very compact form, it involves

binomial coefficients that may cause long runtimes and accuracy problems for large values of upper indices.

The first is the right-circulant determinant sequence with

binomial coefficients and the second is the left-circulant determinant sequence with

binomial coefficients.