# Closed form solution for a binomial coefficient relation?

In following, $$x_{n}$$ is a set of given numbers, n = 0, 1, 2, ...,
$$y_{n}$$ is defined by the following recursive relation of $$x_{n}$$:

For example:

$${\displaystyle {x_{1}=x_{0}y_{1} }}.$$

$${\displaystyle {x_{2}={\binom {1}{0}}x_{0}y_{2} + {\binom {1}{1}}x_{1}y_{1} }}.$$

$${\displaystyle {x_{3}={\binom {2}{0}}x_{0}y_{3} + {\binom {2}{1}}x_{1}y_{2} + {\binom {2}{2}}x_{2}y_{1} }}.$$

For simplicity, we can assume $$x_{0} = 1$$.

Q1: Is there an explicit solution of $$y_{n}$$ in term of $$x_{n}$$ ?

Q2: I assume that above relation should be well known, is it a name for such relation ?

Thank you.

Concerning Q1, under natural general conditions we can express the exponential generating function of $$(y_{j+1})_{j=0}^\infty$$ in terms of the exponential generating function of $$(x_i)_{i=0}^\infty$$.
Indeed, let $$u_i:=\frac{x_i}{i!},\quad v_i:=\frac{y_{i+1}}{i!}$$ for $$i=0,1,\dots$$. Then $$(m+1)u_{m+1}=\sum_{i=0}^m u_i v_{m-i} \tag{1}$$ for $$m=0,1,\dots$$. So, if e.g. $$|v_i|\le C^i$$ for some real $$C\ge1$$ and all $$i$$, then, by induction, $$|u_i|\le C^i$$ for all $$i$$. Hence, we can define the exponential generating functions $$U$$ and $$V$$ of $$(y_{j+1})_{j=0}^\infty$$ and $$(x_i)_{i=0}^\infty$$ by $$U(s):=\sum_{i=0}^\infty u_is^i,\quad V(s):=\sum_{i=0}^\infty v_is^i$$ for all $$s$$ close enough to $$0$$. Then (1) can be rewritten as $$U'(s)=U(s)V(s)$$, so that $$V=U'/U.$$
• @david : Indeed, the binomial coefficient does not appear in (1), and that is the key. The factorial factors in the binomial coefficient get absorbed into the $u_i$'s and $v_i$'s. – Iosif Pinelis Nov 20 '19 at 19:27