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How is the number pattern related to the sucessive derivatives of tan(x) formed?

Tn+1(u)= [d(Tn(u))/du] {1+u^2} 

下一個函數 等於 前一個函數 微分 再乘上 (1+ u^2)

tan(x) u
1+tan(x)^2 1+u^2
2*tan(x)+2*tan(x)^3 2*u+2*u^3
2+8*tan(x)^2+6*tan(x)^4 2+8*u^2+6*u^4
16*tan(x)+40*tan(x)^3+24*tan(x)^5 16*u+40*u^3+24*u^5
16+136*tan(x)^2+240*tan(x)^4+120*tan(x)^6
16+136*tan(x)^2+240*tan(x)^4+120*tan(x)^6 		16+136*u^2+240*u^4+120*u^6
272*tan(x)+1232*tan(x)^3+1680*tan(x)^5+720*tan(x)^7...
272*tan(x)+1232*tan(x)^3+1680*tan(x)^5+720*tan(x)^7... 		272*u+1232*u^3+1680*u^5+720*u^7
272+3968*tan(x)^2+12096*tan(x)^4+13440*tan(x)^6+504...
272+3968*tan(x)^2+12096*tan(x)^4+13440*tan(x)^6+504... 		272+3968*u^2+12096*u^4+13440*u^6+5040*u^8
272+3968*u^2+12096*u^4+13440*u^6+5040*u^8
7936*tan(x)+56320*tan(x)^3+129024*tan(x)^5+120960*t...
7936*tan(x)+56320*tan(x)^3+129024*tan(x)^5+120960*t...
7936*tan(x)+56320*tan(x)^3+129024*tan(x)^5+120960*t... 		7936*u+56320*u^3+129024*u^5+120960*u^7+40320*u^9
7936*u+56320*u^3+129024*u^5+120960*u^7+40320*u^9
7936+176896*tan(x)^2+814080*tan(x)^4+1491840*tan(x)...
7936+176896*tan(x)^2+814080*tan(x)^4+1491840*tan(x)...
7936+176896*tan(x)^2+814080*tan(x)^4+1491840*tan(x)... 		7936+176896*u^2+814080*u^4+1491840*u^6+1209600*u^8+...
7936+176896*u^2+814080*u^4+1491840*u^6+1209600*u^8+...