show the formula always gives an integer

$$\frac{(2m)!(2n)!}{m!n!(m+n)!}$$

I don't remember where I read this problem, but it said this can be proved using a simple counting argument (like observing that $\frac{(3m)!}{m!m!m!}$ is just the number of ways of permuting m identical things of type 1, m of type-2 and m of type-3).

I found this paper

I. M. Gessel, G. Xin, A Combinatorial Interpretation of the
Numbers $6(2n!)/n!(n+2)!$, Journal of Integer Sequences 8 (2005) Article 05.2.3

whose abstract says:


  It is well known that the numbers $\frac{(2m)!(2n)!}{m!n!(m+n)!}$ are integers, but in
  general there is no known combinatorial interpretation for them. When $m = 0$ these
  numbers are the middle binomial coefficients ${2n \choose n}$, and when $m = 1$ they are twice the Catalan numbers. In this paper, we give combinatorial interpretations for these
  numbers when $m = 2$ or $3$.


According to the authors, the first appearance of the problem of whether it's always an integer is in this note by Catalan:

E. Catalan, Question 1135, Nouvelles Annales de Mathematiques 13 (1874), 207.

Barry's comments to this post below point out an incorrect solution to the problem in a more general form was given by Bourguet here:

M. L. Barbier, Solutions des questions proposées dans les Nouvelles annales, Nouvelles Annales de Mathematiques 14 (1875) 66-92,

and the error was pointed out (and corrected?) by Catalan here:

E. Catalan, Correspondance, Nouvelles Annales de Mathematiques 14 (1875) 178-180.

Also in a comment to this post, Karan gave a link to a paper that proves the numbers in question are integers by a recurrence relation:

D. Callan, A Combinatorial Interpretation for a
Super-Catalan Recurrence, Journal of Integer Sequences, 8 (2005) Article 05.1.8.

Edit: The numbers in question are apparently called super Catalan numbers, and I just found this paper whose abstract says, "we show that the super Catalan numbers are special values of the Krawtchouk polynomials by deriving an expression for the super Catalan numbers in terms of a signed set."

E. Georgiadis, A. Munemasa, H. Tanaka, A Note on Super Catalan Numbers, Interdisciplinary Information Sciences, 18 (2012) 23-24.
