Triangular / Taxicab

If $\sum_{n=0}^{\infty}\sum_{k=1}^5\frac{k+1}{n+k}\left(\prod_{k=1}^5(n+k)\right)^{-1}=\frac{T_{a}+1}{\text{Ta}(b)-1}$ where $a,b$ are positive integers and $b$ is prime , then find the value of $a+b+1$ .

Notation : $T_{m\geq 0} ,\text{Ta}(n{\geq 1})$ are mth triangular and nth Taxicab numbers respectively.

While entering your answer count $T_0=0$ as 1st triangular number.

Original problem