Cholesky decomposition of $I_{n\times n}-\frac{1}{n+x}\iota_{n}\iota_{n}^{T}$

Question

I need to compute the Cholesky decomposition of the following matrix:

$\varPi=I_{n\times n}-\frac{1}{n+x}\iota_{n}\iota_{n}^{T}$

Here $n$ is the dimension of the matrix and $x>0$. $\iota_{n}$ is an $n\times1$ column vector. The matrix is guaranteed to be positive definite.

Due to this simple structure is there anyway to obtain an analytical expression for the Cholesky decomposition or at least simplify the computational burden?