A199205 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.

1, 1, 2, 4, 9, 17, 30, 50, 77, 113, 156, 212, 279, 355, 447, 560, 684, 822, 985, 1171, 1375, 1599, 1856, 2134, 2445, 2769, 3125, 3519, 3939, 4376, 4857, 5372, 5914, 6484, 7083, 7717, 8411, 9130, 9882, 10683, 11524, 12393

Offset

1,3

Links

Table of n, a(n) for n=1..42.

Example

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.

Maple

f:= proc(n) option remember;
      `if`(n=1, {[0, 0, 0]},
                {seq(seq(seq( [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)})
    end:
a:= n-> nops(map(x-> x[3], f(n))):
seq(a(n), n=1..20);

Mathematica

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}]}];
a[n_] := Length @ Union @ (#[[3]]& /@ f[n]);
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 32}] (* Jean-François Alcover, Jun 08 2018, after Alois P. Heinz *)

Crossrefs

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.

Sequence in context: A207813 A136379 A065026 * A192967 A182806 A302832

Adjacent sequences: A199202 A199203 A199204 * A199206 A199207 A199208

Keyword

nonn

Author

Alois P. Heinz, Nov 03 2011

Extensions

a(41)-a(42) from Alois P. Heinz, Jun 01 2015

Status

approved