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 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. 10
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 LINKS 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

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Last modified June 6 01:16 EDT 2020. Contains 334858 sequences. (Running on oeis4.)