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A293239 Number of terms in the fully expanded n-th derivative of x^x.

%I A293239 #15 Oct 06 2017 02:05:19
%S A293239 1,2,4,7,11,15,21,28,35,43,53,64,76,88,102,117,133,149,167,186,206,
%T A293239 226,248,271,295,319,345,372,400,428,458,489,521,553,587,622,658,694,
%U A293239 732,771,811,851,893,936,980,1024,1070,1117,1165,1213,1263,1314,1366,1418
%N A293239 Number of terms in the fully expanded n-th derivative of x^x.
%C A293239 Conjecture: the 2nd differences are eventually periodic: 1, 1, 1, 0, 2, 1, 0, 1, [2, 1, 1, 0].
%H A293239 Vaclav Kotesovec, <a href="/A293239/b293239.txt">Table of n, a(n) for n = 0..500</a>
%F A293239 Conjecture: a(n) ~ n^2/2. - _Vaclav Kotesovec_, Oct 05 2017
%F A293239 Conjectures from _Colin Barker_, Oct 05 2017: (Start)
%F A293239 G.f.: (1 + x^2 + x^3 + x^6 - x^8 + x^9 + x^12 - x^13) / ((1 - x)^2*(1 - x^4)).
%F A293239 a(n) = (5 + (-1)^n + (1-i)*(-i)^n + (1+i)*i^n + 2*n + 4*n^2) / 8 for n>7 where i=sqrt(-1).
%F A293239 a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6) for n>6.
%F A293239 (End)
%e A293239 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.
%t A293239 Join[{1}, Length /@ Rest[NestList[Expand[D[#, x]] &, x^x, 53]]]
%Y A293239 Cf. A281434.
%K A293239 nonn
%O A293239 0,2
%A A293239 _Vladimir Reshetnikov_, Oct 03 2017