While analyzing some parallel-computing related algorithm, I came across a probability distribution with a particularly nice property (at least to me), but I am unable to write it down explicitly. 


Let $X$ be a r.v., let $F\_{X}(s)$ be its p.g.f., and let
$$B\_{d}(s)=\sum\_{i=0}^{d}\binom{d}{i}\left(\frac{1}{d}\right)^{i}\left(1-\frac{1}{d}\right)^{d-i}s^{i}=\left(1-\frac{1}{d}+\frac{s}{d}\right)^{d}$$
 be a p.g.f. of $\left(d,\frac{1}{d}\right)$-binomial distributed r.v. $b\_{d}$, for any integer $d>1$. The distribution in question may be defined as 
$$F\_{X}\left(s\right)=s\sum\_{i=0}^{d}\binom{d}{i}\left(\frac{1}{d}\right)^{i}\left(1-\frac{1}{d}\right)^{d-i}F\_{X}^{i}\left(s\right)=s\left(1-\frac{1}{d}+\frac{F\_{X}\left(s\right)}{d}\right)^{d}.$$


The r.v. $X$ may be intuitively described as follows: A student arrives into town with $1$, and earns an additional $b\_{d}$ each month. At the end of each month, she also pays a rent of $1$. The r.v. $X$ basically measures the time before our student is thrown out of the apartment.


If at some time the balance of our student is $k$, we expect her to stay for an additional $\sum\_{i=1}^{k}X$ months (and this is independent of her past affairs). Also, the p.g.f. of $p$-geometric distribution $G\_{p}$ is memoryless by the fact that 


$$G\_{p}(s)=s\cdot\text{Ber}\_{p}\left(G\_{p}\left(s\right)\right),$$


where $\text{Ber}\_{p}$ is p.g.f. of $p$-Bernulli distributed r.v.. Similarly, in our case we can clearly write 
$F\_{X}$ as


$$F\_{X}\left(s\right)=s\cdot B\_{d}\left(F\_{X}\left(s\right)\right).$$


So far, I am only able to calculate (by derivation of $F\_X$) that $\mathbb{E}(X)=\infty$, and also the first few coefficients of $F\_X$, but I would be very interested in the precise description of $F\_{X}$, either in a closed-form of p.g.f., or its coefficients, or cumulative distribution function of $X$.



Total progeny of a Galton-Watson branching process - standard textbook question
The answer above is fine, nevertheless I make some hopefully useful supplementary remarks (the first two essentially reformulating Did's answer)


(1) It is well known (see e.g. Feller I, 3rd ed., p.299) that the generating function $r(s)$ of the total progeny in a Galton-Watson process (started with one individual)
 with reproduction function $h(s)$ is 
the unique positive solution of the implicit equation
 $$r(s)=s h(r(s))\;\;\;.$$
Your $X$ thus describes the total progeny of a GW-process with binomial reproduction $b\_d(s)=(q+ps)^d$ in the ''critical'' case $p=\frac{1}{d}$.


(2) For $h(s)=b\_d(s)$ the coefficients of $F\_X(s)$ are (use Lagrange-inversion) $[s^0] F\_X(s)=0$ and for $n\geq 1$


$$ [s^n] F\_X(s)= \frac{1}{n} [t^{n-1}] (q+pt)^{nd}= \frac{1}{n} {nd \choose n-1} p^{n-1} q^{n(d-1)+1}=\frac{1}{nd+1}{nd+1 \choose n} p^{n-1} q^{n(d-1)+1}$$ 


(This is a (special) case of the **Otter-Dwass** formula.)


(3) The $d$-ary tree function $T\_d(z)=\sum\_{n\geq 0} \frac{1}{n(d-1)+1}{nd \choose n} z^n$ is the (formal) solution of the equation
$$T\_d(z)=1+z(T\_d(z))^d$$
$T\_d$ is known from combinatorics: the $n$-th coefficient of $T\_d$ is the no. of $d-$ary trees with $n$ nodes. 
$F\_X$ is related to $T\_d$ as follows:
 $$F\_X(s)=s\left(qT\_d(spq^{d-1})\right)^d$$
Thus $X$ is distributed as $1+\sum\_{i=1}^d Y\_i$, where the $Y\_i$ are iid with
g.f. $g\_Y(s)=qT\_d(spq^{d-1})$


