A数列計算表

A数列の計算表です。間違いなどありましたら教えて下さい。
$a\in \mathbb{N}$について$(a^n)=(\underset{n個}{\underbrace{a,...,a}})$です
$\uparrow$はクヌースの矢印表記です。
$A$ 数列内の数字$a$について$a\underset{n個}{\underbrace{,...,}}a$ を$a^n$ で略記し、
数字の列$s$と数字$b$について$(s,b\underset{n個}{\underbrace{,..,}}b)$を$(s,b{)}^n$で略記しています。
$n$は十分大きな数とします。
$$\begin{array}{rl}()[0]& =0\\ (0)[0]& =()[0+1]\\ & =()[1]\\ & =1\\ (0,0)[0]& =(0)[1]\\ & =()[1+1]\\ & =2\\ (0^n)[0]& =()[n]\\ & =n\\ (0^n)[n]& =()[n+n]\\ & =2n\\ (1,1)[n]& =(1,0{)}^n[n]\\ & =(1,0^n)[n]\\ & =(0^{2n})[2n]\\ & =2^{2}n\\ & =4n\\ (1,1,1)[n]& =(1,1,0^n)[n]\\ & =(1,1)[2n]\\ & =(1,0^{2n})[2n]\\ & =(1)[4n]\\ & =(0^{4n})[4n]\\ & =2^{3}n\\ & =8n\\ (1^n)[n]& =()[2^{n}n]\\ & =2^{n}n >2\uparrow n\\ (2)[n]& =(1^n)[n] >2\uparrow n\\ (2,1)[n]& =(2,0^n)[n]\\ & =(2)[2n]\\ & =(1^{2n})[2n]\\ & =2^{2n}\cdot 2n>2\uparrow 2n\\ (2,1,1)[n]& =(2,1,0^n)[n]\\ & =(2,1)[2n]\\ & =(2,0^{2n})[2n]\\ & =(2)[4n]\\ & =(1^{4n})[4n]\\ & =2^{2^{2}n}\cdot 2^{2}n>2\uparrow 2\uparrow n\\ (2,1^n)[n]& >\underset{n個}{\underbrace{2\uparrow \cdots \uparrow 2}}\uparrow n\\ & >2\uparrow \uparrow n\uparrow n\\ & \approx 2\uparrow \uparrow n\\ (2,2)[n]& =(2,1^n)[n] >2\uparrow \uparrow n\\ (2,2,1)[n]& =(2,2,0^n)[n]\\ & =(2,2)[2n]>2\uparrow \uparrow 2n\\ (2,2,1,1)[n]& =(2,2,1,0^n)[n]\\ & =(2,2,1)[2n]\\ & >2\uparrow \uparrow (2\uparrow 2)n\\ (2,2,1^n)[n]& >2\uparrow \uparrow (\underset{n個}{\underbrace{2\uparrow \cdots \uparrow 2}})n\\ & >2\uparrow \uparrow (2\uparrow \uparrow n)n\\ & >2\uparrow \uparrow (2\uparrow \uparrow n)\\ (2,2,2)[n]& =(2,2,1^n)[n] >2\uparrow \uparrow (2\uparrow \uparrow n+1)\\ (2,2,2,2)[n]& =(2,2,2,1^n)[n]>2\uparrow \uparrow (2\uparrow \uparrow (2\uparrow \uparrow n+1))\\ (2^n)[n]& >\underset{n個}{\underbrace{2\uparrow \uparrow \cdots \uparrow \uparrow 2}}\uparrow \uparrow n\\ & \approx 2\uparrow \uparrow \uparrow n=2{\uparrow }^{3}n\\ (3)[n]& =(2^n)[n]>2{\uparrow }^{3}n\\ (3,1)[n]& =(3)[2n]>2{\uparrow }^{3}2n\\ (3,1^n)[n]& =(3)[2\uparrow n]>2{\uparrow }^3(2\uparrow n)\\ (3,2)[n]& =(3,1^n)[n]>2{\uparrow }^3(2\uparrow n)\\ (3,2,2)[n]& =(3,2)[2\uparrow n]>2{\uparrow }^3(2\uparrow 2\uparrow n)\\ (3,2^n)[n]& >2{\uparrow }^3(2{\uparrow }^{2}n)+2>2{\uparrow }^3(2{\uparrow }^{2}n)\\ (3,3)[n]& =(3,2^n)[n]>2{\uparrow }^3(2{\uparrow }^{2}n)\\ (3,3,3)[n]& =(3,3,2^n)[n]\\ & >(3,3)[2{\uparrow }^{3}n]\\ & >2{\uparrow }^3(2{\uparrow }^{2}2{\uparrow }^{3}n)\\ & >2{\uparrow }^{3}2{\uparrow }^{3}n\\ (4)[n]& =(3^n)[n]\\ & >\underset{n個}{\underbrace{2{\uparrow }^3\cdots {\uparrow }^{3}2}}{\uparrow }^{3}n\\ & \approx 2{\uparrow }^{4}n\\ (5)[n]& >2{\uparrow }^{5}n\\ (m)[n]& >2{\uparrow }^{m}n\\ (n)[n]& >2{\uparrow }^{n}n\approx f_{\omega }(n)\end{array}$$
また$A(n,n)$をアッカーマン関数として
$(n)[n]>A(n,n)$