Let $\{ a\_{k,m} \}$ be a doublely indexed sequence of positive numbers satisfying:  

$a\_{1,n}\leq \frac{1}{n+1}\quad $
and $\quad a\_{k,m} \leq \frac{1}{m+1}(a\_{k-1,m+1}+L a\_{k-1,m+2})\quad \quad (1)$ ,  

where $L$ is a positive constant and indices $k,m,n$ are non-negative integers.

My question is:
what is the upper bound for $a\_{n,0}$ in terms of n ? 

I try to find the bound by induction using relation in $(1)$ 
to reduce the index $n$ in $a\_{n,0}$ to $1$,
but I'm not able to compute the final representation in terms of $a\_{1,k} (k\geq 1)$.