Let's operator $ \overset {k}{\underset{i =n}{ \lower 3pt { \LARGE\Xi}}}a_i $ means to add all possible products $a_i$ in such way, that sum of their indexs equal to n and the amount of elements of that produkt is equal to k.
eqamples
$ \overset {k=3}{\underset{i =6}{ \lower 3pt { \LARGE\Xi}}}a_i =a_1 a_1 a_4 +a_1 a_4 a_1 +a_4 a_1 a_1+a_3 a_2 a_1+a_3 a_1 a_2+ a_1 a_3 a_2 +a_2 a_3 a_1 +a_1 a_2 a_3+a_2 a_1 a_3 =3 a_1^2 a_4+6a_1 a_2 a_3$
$ \overset { 1<k<5}{\underset{i =4}{ \lower 3pt { \LARGE\Xi}}}a_i =a_1 a_3+a_3 a_1+a_2 a_2+a_1 a_1 a_2+ a_1 a_2 a_1 + a_2 a_1 a_1+a_1 a_1 a_1 a_1=2 a_1 a_3+a_2^2 +3a_1^2a_2+a_1^4$
$ \overset {k=1}{\underset{i =9}{ \lower 3pt { \LARGE\Xi}}}a_i =a_9$
Is it matematicly possible to give explicit formula of such recursive equation?
$a_m(n+1)= \overset {0<k<m}{\underset{i = (m-1)}{ \lower 3pt { \LARGE\Xi}}}\frac{a_i(n)}{k!} $.
For $m=1$, $a_1(n+1)=1$ (It is not clear what $\overset {0<k<1}{\underset{i = (0)}{ \lower 3pt { \LARGE\Xi}}}\frac{a_i(n)}{k!} $ means, but from solution this is the interpretation.
Im asking because to be honest im not truelly understand when recursive equation has explicite formula. For example there exsists excplicit formula for Fibonacci numbers but I don't know how show something like that.