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.
%I A290268 #26 Oct 10 2017 15:24:57
%S A290268 1,2,5,8,13,18,25,31,41,49,61,71,85,97,113,126,145,160,181,198,221,
%T A290268 240,265,285,313,335,365,389,421,447,481,508,545,574,613,644,685,718,
%U A290268 761,795,841,877,925,963,1013,1053,1105,1146,1201,1244,1301,1346,1405,1452
%N A290268 Number of terms in the fully expanded n-th derivative of x^(x^2).
%F A290268 Conjectured g.f.: (1 + x + 2*x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 2*x^6 + x^7 + 2*x^8 + x^9)/((1 - x)*(1 - x^2)*(1 - x^8)).
%F A290268 Conjecture: a(n) = (8*n^2 + 15*n + 14 + (n + 2)*(-1)^n + (2 - 4*sqrt(2)*sin(Pi*n/4))*sin(Pi*n/2))/16.
%F A290268 From _Peter Luschny_, Oct 09 2017: (Start) Assuming the conjecture:
%F A290268 a(n) = n^2/2 + n + 1 - (n mod 2)*(1/2 + floor((n + 1)/8)).
%F A290268 a(2*n) = A001844(n) and a(2*n + 1) = A293296(n + 1).
%F A290268 Signature of the linear recurrence: {0, 2, 0, -1, 0, 0, 0, 1, 0, -2, 0, 1}. (End)
%e A290268 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.
%p A290268 a := n -> `if`(n=0, 1, nops(expand(diff(x^(x^2), x$n)))):
%p A290268 seq(a(n), n = 0..30); # _Peter Luschny_, Oct 08 2017
%t A290268 Join[{1}, Length /@ Rest[NestList[Expand[D[#, x]] &, x^x^2, 53]]]
%t A290268 (* Use it only to check the conjecture, not to compute the values: *)
%t A290268 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 *)
%Y A290268 Cf. A001844, A281434, A293239, A293296.
%K A290268 nonn
%O A290268 0,2
%A A290268 _Vladimir Reshetnikov_, Oct 06 2017