%I #10 Oct 04 2014 10:45:39

%S 1,1,2,4,9,20,48,115,286,717,1815,4574,11505,28546,69705,166010

%N Number of distinct values taken by 8th 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 8th derivative at x=1: (x^(x^(x^x))) -> 269128; ((x^x)^(x^x)), ((x^(x^x))^x) -> 382520; (x^((x^x)^x)) -> 511216; (((x^x)^x)^x) -> 646272.

%p # load programs from A215703, then:

%p a:= n-> nops({map(f-> 8!*coeff(series(subs(x=x+1, f),

%p x, 9), x, 8), T(n))[]}):

%p seq(a(n), n=1..10);

%Y Cf. A000081 (distinct functions), A000108 (parenthesizations), A000012 (first derivatives), A028310 (2nd derivatives), A199085 (3rd derivatives), A199205 (4th derivatives), A199296 (5th derivatives), A199883 (6th derivatives), A002845, A003018, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A082499, A196244, A198683, A215703, A215838. Column k=8 of A216368.

%K nonn,more

%O 1,3

%A _Alois P. Heinz_, Aug 29 2012