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Topic: Proving that a ratio of factorials is an integer
Replies: 2   Last Post: Dec 15, 2006 10:06 AM

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tchow@lsa.umich.edu

Posts: 1,133
Registered: 12/6/04
Proving that a ratio of factorials is an integer
Posted: Dec 14, 2006 4:26 PM
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In Problem 6514 of the American Mathematical Monthly, Richard Askey asked
for a proof that (3m+3n)!(3n)!(2m)!(2n)!/(2m+3n)!(m+2n)!(m+n)!m!n!n! is
always an integer if m and n are nonnegative integers. The solution by
Gregg Patruno gave a general method for attacking problems of this type.
Reference: Amer. Math. Monthly 94 (1987), 1012-1014, or see JSTOR:

http://www.jstor.org/view/00029890/di991727/99p0111d/0

What I'm wondering is, is there a theorem of the form, "Whenever an
expression like this is always an integer, then there is always an
expression for it in terms of binomial coefficients and polynomials
in m and n that makes it obvious that it is an integer"? For a slightly
different example of the kind of thing I'm after, the Catalan number
(2n)!/n!(n+1)! is not obviously an integer when you write it in that
form, but (2n)!/n!(n+1)! = (2n choose n) - (2n choose n-1), which is
obviously an integer. Patruno's method does not seem to answer my
question directly.
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences




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