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%I #33 Jun 08 2018 03:37:24
%S 1,1,2,4,9,17,30,50,77,113,156,212,279,355,447,560,684,822,985,1171,
%T 1375,1599,1856,2134,2445,2769,3125,3519,3939,4376,4857,5372,5914,
%U 6484,7083,7717,8411,9130,9882,10683,11524,12393
%N Number of distinct values taken by 4th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.
%e a(5) = 9 because the A000108(4) = 14 possible parenthesizations of x^x^x^x^x lead to 9 different values of the 4th derivative at x=1: (x^(x^(x^(x^x)))) -> 56; (x^(x^((x^x)^x))) -> 80; (x^((x^(x^x))^x)), (x^((x^x)^(x^x))) -> 104; ((x^x)^(x^(x^x))), ((x^(x^(x^x)))^x) -> 124; ((x^(x^x))^(x^x)) -> 148; (x^(((x^x)^x)^x)) -> 152; ((x^x)^((x^x)^x)), ((x^((x^x)^x))^x) -> 172; (((x^x)^x)^(x^x)), (((x^(x^x))^x)^x), (((x^x)^(x^x))^x) -> 228; ((((x^x)^x)^x)^x) -> 344.
%p f:= proc(n) option remember;
%p `if`(n=1, {[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 h=f(n-j)), g=f(j)), j=1..n-1)})
%p end:
%p a:= n-> nops(map(x-> x[3], f(n))):
%p seq(a(n), n=1..20);
%t f[n_] := f[n] = If[n == 1, {{0, 0, 0}}, Union @ Flatten[#, 3]& @ {Table[ Table[Table[{2 + g[[1]], 3*(1 + g[[1]] + h[[1]]) + g[[2]], 8 + 12*g[[1]] + 6*h[[1]]*(1 + g[[1]]) + 4*(g[[2]] + h[[2]]) + g[[3]]}, {h, f[n - j]}], {g, f[j]}], {j, 1, n - 1}]}];
%t a[n_] := Length @ Union @ (#[[3]]& /@ f[n]);
%t Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 32}] (* _Jean-François Alcover_, Jun 08 2018, after _Alois P. Heinz_ *)
%Y Cf. A000081 (distinct functions), A000108 (parenthesizations), A000012 (first derivatives), A028310 (2nd derivatives), A199085 (3rd derivatives), A199296 (5th derivatives), A002845, A003018, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A082499, A196244, A198683, A215703, A215834. Column k=4 of A216368.
%K nonn
%O 1,3
%A _Alois P. Heinz_, Nov 03 2011
%E a(41)-a(42) from _Alois P. Heinz_, Jun 01 2015
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