I want to know if there are conjectures similar to this one, I know there is the Bell primes conjecture or Gardner conjecture (mentioned in this page https://en.wikipedia.org/wiki/Bell_number), but this one seems to be different.
A recurrent prime of order $n$ is a prime number of the form $\sum_{i_{n-1}=0}^{i_{n}}\cdots \sum_{i_{0}=0}^{i_{1}} a_{(n-1);i_{n-1}} \cdots a_{(0);i_{0}}$ where $a_{(k);i_{k}}$'s are sequences of natural numbers.
Let $(a_{n})_{n \in \mathbb{N}}$ be a strictly increasing sequence of natural numbers. We define the numbers $R_{a_{n}-a_{0}-n}^{k}$ by:
$$ \begin{cases} R_{0}^{0}=1, & \text{} \\[2ex] R_{a_{n}-a_{0}-n}^{k}=\sum_{i=0}^{k}{(a_{n}-a_{0}-n-k+1)}^{(k-i)}R_{a_{n-1}-a_{0}-(n-1)}^{i} \label{1}, & \text{$\forall\ 0\leq k\leq a_{n}-a_{0}-n$} & (1)\\[2ex] R_{a_{n}-a_{0}-n}^{k}=0, & \text{$\forall\ 0\leq a_{n}-a_{0}-n < k$} \end{cases} $$
where ${x}^{(n)}$ is the rising factorial.
If we substitute repeatidly in $(1)$ we find that:
$$R_{a_{n}-a_{0}-n}^{k}=\sum_{i_{n-1}=0}^{k}{(a_{n}-a_{0}-n-k+1)}^{(k-i_{n-1})}\sum_{i_{n-2}=0}^{i_{n-1}}{(a_{n-1}-a_{0}-(n-1)-i_{n-1}+1)}^{(i_{n-1}-i_{n-2})}\cdots\sum_{i_{0}=0}^{i_{1}}{(a_{1}-a_{0}-i_{1})}^{(i_{1}-i_{0})}R_{0}^{i_{0}}$$
$R_{0}^{i_{0}}=\delta_{i_{0}}=\mathrm{Kronecker\ delta}$, so :
$$R_{a_{n}-a_{0}-n}^{k}=\sum_{i_{n-1}=0}^{k}{(a_{n}-a_{0}-n-k+1)}^{(k-i_{n-1})}\sum_{i_{n-2}=0}^{i_{n-1}}{(a_{n-1}-a_{0}-(n-1)-i_{n-1}+1)}^{(i_{n-1}-i_{n-2})}\cdots\sum_{i_{0}=0}^{i_{1}}{(a_{1}-a_{0}-i_{1})}^{(i_{1}-i_{0})}\delta_{i_{0}}$$
After doing some calculations and number triangles we find that:
If $a_{n}=2n$ then:
$$ \begin{split} 2&=\sum_{k=0}^{1}R_{1}^{k}\\ 7&=\sum_{k=0}^{2}R_{2}^{k}\\ 37&=\sum_{k=0}^{3}R_{3}^{k}\\ \end{split} $$
If $a_{n}=3n$ then: $$ \begin{split} 5&=\sum_{k=0}^{2}R_{2}^{k}\\ 107&=\sum_{k=0}^{4}R_{4}^{k}\\ 6053&=\sum_{k=0}^{6}R_{6}^{k}\\ \end{split} $$
If $a_{n}=4n$ then: $$ \begin{split} 3\,389\,929&=\sum_{k=0}^{9}R_{9}^{k}=\sum_{i_{3}=0}^{9}1\sum_{i_{2}=0}^{i_{3}}{(10-i_{3})}^{(i_{3}-i_{2})}\sum_{i_{1}=0}^{i_{2}}{(7-i_{2})}^{(i_{2}-i_{1})}\sum_{i_{0}=0}^{i_{1}}{(4-i_{1})}^{(i_{1}-i_{0})}\delta_{i_{0}}\\ \end{split} $$
these numbers are particular recurrent primes.
also if $a_{n}=5n$ we have:
$$ \begin{split} 197\,269&=\sum_{k=0}^{8}R_{8}^{k}\\ \end{split} $$
another example, if $a_{n}=n^{2}$ then:
$$ \begin{split} 2\,999&=\sum_{k=0}^{6}R_{6}^{k}\\ 2\,883\,135\,407&=\sum_{k=0}^{12}R_{12}^{k}\\ \end{split} $$
We state the following conjecture :
$\textbf{Conjecture}$
Let $(a_{n})_{n \in \mathbb{N}}$ be a strictly increasing sequence of natural numbers.
There are infinitely many pairs $(N, p)$ such that:
$$p=\sum_{k=0}^{a_{N}-a_{0}-N}R_{a_{N}-a_{0}-N}^{k}$$
where $N$ is a natural number, and $p$ a prime number.
Also what is the explicit formula of the numbers $R_{a_{n}-a_{0}-n}^{k}$? I know when $a_{n}=2n$ we have:
$$R_{a_{n}-a_{0}-n}^{k}=R_{n}^{k}=\frac{(n+k)!}{2^{k}k!(n-k)!}$$
I found that by drawing the number triangle of $R_{n}^{k}$ and comparing it with this one https://oeis.org/A001498, but in general I don't know how to proceed, it seems very hard.
