cp's OEIS Frontend

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.

%I A199205 #33 Jun 08 2018 03:37:24
%S A199205 1,1,2,4,9,17,30,50,77,113,156,212,279,355,447,560,684,822,985,1171,
%T A199205 1375,1599,1856,2134,2445,2769,3125,3519,3939,4376,4857,5372,5914,
%U A199205 6484,7083,7717,8411,9130,9882,10683,11524,12393
%N 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.
%e A199205 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 A199205 f:= proc(n) option remember;
%p A199205       `if`(n=1, {[0, 0, 0]},
%p A199205                 {seq(seq(seq( [2+g[1], 3*(1 +g[1] +h[1]) +g[2],
%p A199205                  8 +12*g[1] +6*h[1]*(1+g[1]) +4*(g[2]+h[2])+g[3]],
%p A199205                  h=f(n-j)), g=f(j)), j=1..n-1)})
%p A199205     end:
%p A199205 a:= n-> nops(map(x-> x[3], f(n))):
%p A199205 seq(a(n), n=1..20);
%t A199205 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 A199205 a[n_] := Length @ Union @ (#[[3]]& /@ f[n]);
%t A199205 Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 32}] (* _Jean-François Alcover_, Jun 08 2018, after _Alois P. Heinz_ *)
%Y A199205 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 A199205 nonn
%O A199205 1,3
%A A199205 _Alois P. Heinz_, Nov 03 2011
%E A199205 a(41)-a(42) from _Alois P. Heinz_, Jun 01 2015