Let \(a_1=b_1=1\) and \(a_n=a_{n-1}+(n-1), b_n=b_{n-1}+\) \(a_{n-1}, \forall n \geq 2\). If \(S=\sum_{n=1}^{10} \frac{b_…

Let \(a_1=b_1=1\) and \(a_n=a_{n-1}+(n-1), b_n=b_{n-1}+\) \(a_{n-1}, \forall n \geq 2\). If \(S=\sum_{n=1}^{10} \frac{b_n}{2^n}\) and \(T=\sum_{n=1}^8 \frac{n}{2^{n-1}}\), then \(2^7(2 S-T)\) is equal to ______ .