Let ${\mathrm{A}}_{\mathrm{n}},(\mathrm{n}\in \mathrm{N})$be a matrix of order $(2\mathrm{n}-1)\times (2\mathrm{n}-1)$,such that ${\mathrm{a}}_{\mathrm{ij}}=0\mathrm{\forall }\mathrm{i}\ne \mathrm{j}$and ${\mathrm{a}}_{\mathrm{ij}}={\mathrm{n}}^2+\mathrm{i}+1-2\mathrm{n},\mathrm{\forall }\mathrm{i}=\mathrm{j}$where ${\mathrm{a}}_{\mathrm{ij}}$ denotes the element of i^th row and j^th column of $A_n$.

 Let $T_n={\left(-1\right)}^n·\left(sum of all the elements of A_n\right)$ 

Find the value of$\left[\frac{\sum _{\mathrm{n}=1}^{102} {\mathrm{T}}_{\mathrm{n}}}{520200}\right]$, where [.] represents the greatest integer function.