Recognizing a measure whose moments are the motzkin numbers The Catalan numbers are the moments of the 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 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?
 A: Motzkin numbers are a very popular sequence. A lot of identities and formulas are already recorded at OEIS. The analogous integral representation for Motzkin numbers is 
$$M_n=\frac{1}{2\pi} \int_{-1}^3 x^n\sqrt{(3-x)(1+x)}dx.$$

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 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.
A: The formula for the measure of the Motzkin numbers as stated by Gjergij follows from the formula for the Catalan numbers if we write the formula for the Catalan numbers in the form
$\frac{1}{{2\pi }}\int_{ - 2}^2 {x^{2n} \sqrt {4 - x^2 } } dx = C_n $ and observe that the corresponding orthogonal polynomials are the Fibonacci polynomials  $F_{n + 1} (x, - 1)$  defined by $F_n (x, - 1) = xF_{n - 1} (x, - 1) - F_{n - 2} (x, - 1)$  with initial values $F_0 (x, - 1) = 0$ and $F_1 (x, - 1) = 1.$
For the Motzkin numbers the corresponding orthogonal polynomials are $F_{n + 1} (x - 1, - 1).$ Therefore a simple transformation of the integral gives the result.
Edit:
More generally (as answer to the comments by Gjergji and Brendan): 
Let $f(z) = \sum {r(n,c,d)z^n } $ satisfy $f(z) = 1 + czf(z) + d^2 z^2 f(z)^2. $
Then 
$r(n,c,d) = \frac{1}{{2\pi d^2 }}\int_{c – 2d}^{c + 2d} {x^n \sqrt {4d^2  - (x - c)^2 } } dx.$
