%I #15 Sep 26 2014 16:34:56

%S 1,1,2,4,9,20,48,113,262,591,1263,2505,4764,8479,14285,22871,35316,

%T 52755,76517,107826,148914,202715,270622

%N Number of distinct values taken by 6th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.

%e a(4) = 4 because the A000108(3) = 5 possible parenthesizations of x^x^x^x lead to 4 different values of the 6th derivative at x=1: (x^(x^(x^x))) -> 2934; ((x^x)^(x^x)), ((x^(x^x))^x) -> 4908; (x^((x^x)^x)) -> 5034; (((x^x)^x)^x) -> 8322.

%p f:= proc(n) option remember;

%p `if`(n=1, {[0, 0, 0, 0, 0]},

%p {seq(seq(seq([2+g[1], 3*(1 +g[1] +h[1]) +g[2],

%p 8 +12*g[1] +6*h[1]*(1+g[1]) +4*(g[2]+h[2])+g[3],

%p 10+50*h[1]+10*h[2]+5*h[3]+(30+30*h[1]+10*h[2]

%p +15*g[1])*g[1]+(20+10*h[1])*g[2]+5*g[3]+g[4],

%p 45*h[1]*g[1]^2+(120+60*h[2]+15*h[3]+60*g[2]+

%p 270*h[1])*g[1]+54+15*h[3]+30*g[3]+6*g[4]+

%p 60*h[1]*g[2]+15*h[1]*g[3]+30*h[1]+ 20*h[2]*g[2]+

%p 100*h[2]+90*h[1]^2+g[5]+60*g[2]+6*h[4]],

%p h=f(n-j)), g=f(j)), j=1..n-1)})

%p end:

%p a:= n-> nops(map(x-> x[5], f(n))):

%p seq(a(n), n=1..15);

%Y Cf. A000081 (distinct functions), A000108 (parenthesizations), A000012 (first derivatives), A028310 (2nd derivatives), A199085 (3rd derivatives), A199205 (4th derivatives), A199296 (5th derivatives), A002845, A003018, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A082499, A196244, A198683, A215703, A215836. Column k=6 of A216368.

%K nonn,more

%O 1,3

%A _Alois P. Heinz_, Nov 11 2011

%E a(22)-a(23) from _Alois P. Heinz_, Sep 26 2014