On entire solutions with a two-member recurrent formula for Taylor's coefficients of linear differential equations

It is proved that the differential equation $$z^nw^{(n)}+(a_1^{(n-1)}z+a_2^{(n-1)})z^{n-1}w^{(n-1)}+ sum_{k=0}^{n-2}{(a_{n-1-k}^{(k)}z^2+a_{n-k}^{(k)}z+a_{n+1-k}^{(k)})z^kw^{(k)}}=0$$ has an entire solution $f$ with a two-member recurrent formula for its Taylor's coefficients. The growth of such function $f$ is studied. The conditions for coefficients $a_k^{(j)}$ are obtained, under which the solution $f$ is convex or close-to-convex in $mathbb{D}={z:|z|