Binomial sums related to rational approximations to $\zeta (4)$

For the solution $\{u_n{\}}_{n=0}^{\mathrm{\infty }}$ to the polynomial recursion $(n+1{)}^5{u}_{n+1}-3(2n+1)(3n^2+3n+1)(15n^2+15n+4)u_n-3n^3(3n-1)(3n+1)u_{n-1}=0$, where $n=1,2,\mathrm{.}\mathrm{.}\mathrm{.}$, with the initial data $u_0=1$, $u_1=12$, we prove that all $u_n$ are integers. The numbers $u_n$, $n=0,1,2,\mathrm{.}\mathrm{.}\mathrm{.}$, are denominators of rational approximations to $\zeta (4)$ (see math.NT/0201024). We use Andrews’s generalization of Whipple’s transformation of a terminating ${}_7{F}_6(1)$-series and the method from math.NT/0311114.