This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
With[{m = 20}, Table[(4 n + 3 n^2 + 2 n^3)/3, {n, m}] - Length /@ Rest[NestList[Expand[D[#, x]] &, x^x^x, m]]]
For n = 3, the 3rd derivative of x^x is x^x + 3*x^x*log(x) + 3*x^x*log^2(x) + x^x*log^3(x) + 3*x^(x-1) + 3*x^(x-1)*log(x) - x^(x-2), so a(3) = 7.
Join[{1}, Length /@ Rest[NestList[Expand[D[#, x]] &, x^x, 53]]]
For n = 2, the 2nd derivative of x^(x^2) is 3*x^(x^2) + 2*x^(x^2)*log(x) + x^(x^2+2) + 4*x^(x^2+2)*log(x) + 4*x^(x^2+2)*log^2(x), so a(2) = 5.
a := n -> `if`(n=0, 1, nops(expand(diff(x^(x^2), x$n)))): seq(a(n), n = 0..30); # Peter Luschny, Oct 08 2017
Join[{1}, Length /@ Rest[NestList[Expand[D[#, x]] &, x^x^2, 53]]]
(* Use it only to check the conjecture, not to compute the values: *)
LinearRecurrence[{0,2,0,-1,0,0,0,1,0,-2,0,1}, {1,2,5,8,13,18,25,31,41,49,61,71}, 54] (* Peter Luschny, Oct 09 2017 *)
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