%I #26 Mar 06 2019 20:57:21

%S 1,1,2,4,9,20,48,115,283,691,1681,3988,9241,20681,44217,89644

%N Number of distinct values taken by 7th 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 7th derivative at x=1: (x^(x^(x^x))) -> 26054; ((x^x)^(x^x)), ((x^(x^x))^x) -> 41090; (x^((x^x)^x)) -> 47110; (((x^x)^x)^x) -> 70098.

%p T:= proc(n) T(n):=`if`(n=1, [x], map(h-> x^h, g(n-1$2))) end:

%p g:= proc(n, i) option remember; `if`(i=1, [x^n], [seq(seq(

%p seq(mul(T(i)[w[t]-t+1], t=1..j)*v, v=g(n-i*j, i-1)), w=

%p combinat[choose]([$1..nops(T(i))+j-1], j)), j=0..n/i)])

%p end:

%p f:= proc() local i, l; i, l:= 0, []; proc(n) while n>

%p nops(l) do i:= i+1; l:= [l[], T(i)[]] od; l[n] end

%p end():

%p a:= n-> nops({map(f-> 7!*coeff(series(subs(x=x+1, f), x, 8), x, 7), T(n))[]}):

%p seq(a(n), n=1..12);

%Y Column k=7 of A216368.

%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, A215837.

%K nonn,more

%O 1,3

%A _Alois P. Heinz_, Aug 24 2012