Could anyone provide a reference for the following (sort of) generalization of Catalan numbers: the multinomial coefficient
$$
\binom{2k\_1+3k\_2+4k\_3+...}{k\_1+2k\_2+3k\_3+...,k\_1,k\_2,k\_3,...}
$$
is divisible by $k\_1+2k\_2+3k\_3+...+1$.


Denoting the quotient by $C(k\_1,k\_2,k\_3,...)$, one may call these the multiCatalan numbers. They definitely must appear somewhere in combinatorics, but I could not find any reference.


The reason I am sure these numbers must be known is that, for example: $(-1)^{k\_1+k\_2+...}C(k\_1,k\_2,...)$ is the coefficient at $x\_1^{k\_1}x\_2^{k\_2}\cdots$ of the composition inverse of the formal power series $t+x\_1t^2+x\_2t^3+...$;  

$C(k\_1,k\_2,...)$ is the number of faces of the Stasheff polytope $S\_{k\_1+k\_2+...}$ of shape $S\_1^{k\_1}\times S\_2^{k\_2}\times\cdots$ (here for convenience I have redenoted by $S\_n$ the standard $K\_{n+1}$; so $S\_1$ is a point, $S\_2$ a segment, $S\_3$ a pentagon, etc.);  

hence they also enumerate certain kinds of trees, etc., etc. ...



"MultiCatalan numbers"
An early reference is [W. T. Tutte, The number of planted plane trees with a given partition. Amer. Math. Monthly 71 (1964) 272–277](http://www.jstor.org/stable/2312183).


