Certificates of Integrality for Linear Binomials
1999
Say a sequence is linear binomial if its nth term is a quotient
of (i) a binomial coefficient whose parameters are linear in
n and (ii) a product P(n) of factors linear in n.
An example is the familiar Catalan numbers
(2n choose n)/(n + 1) = {1,2,5,14,42,...}.
This paper gives a simple criterion for a linear
binomial sequence to have bounded denominators (so that the sequence has an
integralizing factor). Roughly speaking, the criterion
is: if and only if P(n)'s linear factors are distinct,
and each appears more often in the "symbolic numerator" of the
binomial coefficient than in its "symbolic denominator".
The proof is
algorithmic; applied to 
, it yields the integer
multiplier "3" along with the identity 
, which
serves as a "Certificate of Integrality".
