递推公式
\[F(n)=\begin{cases}
0&(n=0)& \\
1&(n=1)& \\
2F(n-1)+1 &(n>1)&
\end{cases}
\]
构造生成函数求解
\[\begin{array}{lcl}
G(x)=1 \cdot x^1+3\cdot x^2+7\cdot x^3+15\cdot x^4 +\cdots\\\\
2x\cdot G(x)=\qquad 2\cdot x^2+6\cdot x^3 +14 \cdot x^4\;+\cdots\\\\
G(x)-2x\cdot G(x)=x^1+x^2+x^3+x^4+\cdots\\\\
(1-2x)G(x)=\frac{1}{1-x}-1 \\\\
(1-2x)G(x)=\frac{x}{1-x} \\\\
G(x)=\frac{x}{(1-2x)(1-x)}=\frac{1}{1-2x}-\frac{1}{1-x}\\\\\\
\qquad\;\;=(1+2x+2^2x^2+2^3x^3+\cdots+2^nx^n)-(1+x^1+x^2+\cdots+x^n)\\\\
\qquad\;\;=(2^1-1)x+(2^2-1)x^2+\cdots(2^n-1)x^n\\\\
\Rightarrow F(n)=2^n-1
\end{array}
\]