The [Catalan numbers](http://en.wikipedia.org/wiki/Catalan_number) are the moments of the [Wigner semicircle distribution](http://en.wikipedia.org/wiki/Wigner_semicircle_distribution).  

$$ \frac{1}{2\pi} \int\_{-2}^2 x^{2n} \sqrt{4 - x^2} dx = \frac{1}{n+1} \binom{2n}{n} $$
[Motzkin numbers](http://en.wikipedia.org/wiki/Motzkin_number) enumerate the number of paths from (0,0) to (0,n) in steps of (1,1),(1,-1),(1,0) remaining above the x-axis.


For this particular sequence of numbers 1, 1, 2, 4, 9, 21, 51, 127... satisfying the recursion 
$$ M\_n = \frac{3(n-1)M\_{n-2} +(2n+1)M\_{n-1}}{n+2} $$
Is it possible to reconstruct a measure $\rho(x)$ whose moments $\langle x^n \rangle$ follow that sequence?



Recognizing a measure whose moments are the motzkin numbers
Motzkin numbers are a very popular sequence. A lot of identities and formulas are already recorded at [OEIS](http://oeis.org/A001006). The analogous integral representation for Motzkin numbers is 
$$M\_n=\frac{1}{2\pi} \int\_{-1}^3 x^n\sqrt{(3-x)(1+x)}dx.$$




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A few words about the general picture. There is a well known combinatorial theory of orthogonal polynomials and continued fraction which is closely related to Motzkin paths.


By [Favard's theorem](http://en.wikipedia.org/wiki/Favard%27s_theorem) we know that a sequence of polynomials is a system of orthogonal polynomials with respect to some measure if and only if they satisfy a recurrence for all $n$
$$P\_n=(x-a\_n)P\_{n-1}-b\_nP\_{n-2}$$
Now the generating functions of moments $\sum \mu\_ix^i$ when written as a continued fraction tells us the coefficients $a\_n,b\_n$ and therefore the sequence of orthogonal polynomials. One can also interpret these continued fractions as generating functions for weighted Motzkin paths. This is a theorem of Viennot



> 
> The moments of a sequence of orthogonal polynomials are sums of Motzkin paths of length $n$ where horizontal steps at height $k$ are weighted by $a\_k$ and down steps at height $k$ are weighted by $b\_k$.
> 
> 
> 


As a special case the numbers $M\_n$ appear as moments of the sequence of polynomials which satisfies $P\_n=(x-1)P\_{n-1}-P\_{n-2}$ which are shifted and scaled Chebyshev polynomials of second kind, and the identity above holds. However one can give such a combinatorial context to most known families of orthogonal polynomials.


For the relation between orthogonal polynomials and continued fractions, I would recommend "Orthogonal polynomials and random matrices: a Riemann-Hilbert approach" by P. Deift, it is short and a very enjoyable read. For the combinatorial theory see for example "Combinatorial aspects of continued fractions" by P. Flajolet or Viennot's works on orthogonal polynomials which you can find [here](http://web.mac.com/xgviennot/Xavier_Viennot/polyn%C3%B4mes_orthogonaux.html).


