Hausdorff moment problem for combinatorial numbers of Brown and Tutte: exact solution

We investigate the combinatorial sequences $A(M,n)$ introduced by W. G. Brown (1964) and W. T. Tutte (1980) appearing in enumeration of convex polyhedra. Their formula is $$A(M,n)=\frac{2(2M+3)!}{(M+2)!M!}\frac{(4n+2M+1)!}{n!(3n+2M+3)!}$$ with $n,M=0,1,2,\dots$, and we conceive it as Hausdorff moments, where $M$ is a parameter and $n$ enumerates the moments. We solve exactly the corresponding Hausdorff moment problem: $A(M,n)={\int }_0^R{x}^n{W}_M(x)dx$ on the natural support $(0,R)$, $R=4^4\mathrm{/}{3}^3$, using the method of inverse Mellin transform. We provide explicitly the weight functions $W_M(x)$ in terms of the Meijer G-functions $G_{4,4}^{4,0}$, or equivalently, the generalized hypergeometric functions ${}_3{F}_2$ (for $M=0,1$) and ${}_4{F}_3$ (for $M\ge 2$). For $M=0,1$, we prove that $W_M(x)$ are non-negative and normalizable, thus they are probability distributions. For $M\ge 2$, $W_M(x)$ are signed functions vanishing on the extremities of the support. By rephrasing this problem entirely in terms of Meijer G representations we reveal an integral relation which directly furnishes $W_M(x)$ based on ordinary generating function of $A(M,n)$ as an input. All the results are studied analytically as well as graphically.