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 A199085 Number of distinct values taken by 3rd derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1. 10
 1, 1, 2, 4, 7, 11, 15, 20, 26, 32, 39, 47, 55, 64, 74, 84, 95, 107, 119, 132, 146, 160, 175, 191, 207, 224, 242, 260, 279, 299, 319, 340, 362, 384, 407, 431, 455, 480, 506, 532, 559, 587, 615, 644, 674, 704, 735, 767, 799, 832, 866, 900, 935, 971, 1007, 1044 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Number of distinct values taken by 0th and 1st derivative is 1,1,1,1,1,1,... and by 2nd is 1,1,2,3,4,5,... LINKS FORMULA Conjectured g.f.: (x^6-2*x^4-x^2+x-1)*x/(x^5-2*x^4+x^3-x^2+2*x-1). - Alois P. Heinz, Nov 02 2011 EXAMPLE For n=5 there are 7 distinct values: 9, 15, 18, 21, 24, 33, 48, they are given by 3rd derivatives of the following parenthesizations at x=1: x^(x^((x^x)^x)), x^((x^(x^x))^x), (x^x)^(x^(x^x)), x^(((x^x)^x)^x), (x^(x^x))^(x^x), (((x^(x^x)))^x)^x, (((x^x)^x)^x)^x. So a(5)=7. MAPLE f:= proc(n) option remember;       `if`(n=1, {[0, 0]}, {seq(seq(seq([2+g[1], 3*(1+g[1]+h[1])+g[2]],                            h=f(n-j)), g=f(j)), j=1..n-1)})     end: a:= n-> nops(map(x-> x[2], f(n))): seq(a(n), n=1..40);  # Alois P. Heinz, Nov 03 2011 MATHEMATICA f[1] = {x}; f[n_] := Flatten[Table[Outer[Power, f[k], f[n-k]], {k, n-1}]]; Table[Length[Union[D[f[n], {x, 3}] /. x -> 1]], {n, 1, 8}] (* Reshetnikov *) Table[If[n<3, 1, Floor[(n^2-2)/3]], {n, 1, 100}] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2012 *) CROSSREFS Cf. A000081 (upper bound), A000108, A199205 (4th derivatives), A199296 (5th derivatives), A215703, A215842. Column k=3 of A216368. Sequence in context: A194168 A198759 A078617 * A247184 A025703 A025709 Adjacent sequences:  A199082 A199083 A199084 * A199086 A199087 A199088 KEYWORD nonn AUTHOR Vladimir Reshetnikov, Nov 02 2011 EXTENSIONS a(13)-a(56) from Alois P. Heinz, Nov 02 2011 STATUS approved

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