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Proving that a ratio of factorials is an integer
Posted:
Dec 14, 2006 4:26 PM


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), 10121014, 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 n1), which is obviously an integer. Patruno's method does not seem to answer my question directly.  Tim Chow tchowatalumdotmitdotedu The range of our projectileseven ... the artilleryhowever 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



