In a recent answer Max Alekseyev provided two recurrences of the form mentioned in the title which stay integer for a long time. However, they eventually fail.

QUESTIONIs there any (added:strictly increasing) sequence of positive integers $c_1,c_2,c_3,\dots$ satisfying the relation $$ c_{n+1}=\frac{c_n(c_n+n+d)}n $$ for for all $n\geq 1$ (and some integer constant $d$)?

(As Sam Hopkins notes, it would be also very interesting if the indices of the sequence start from some $k>1$ rather than from 1.)