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 A293239 Number of terms in the fully expanded n-th derivative of x^x. 4

%I #15 Oct 06 2017 02:05:19

%S 1,2,4,7,11,15,21,28,35,43,53,64,76,88,102,117,133,149,167,186,206,

%T 226,248,271,295,319,345,372,400,428,458,489,521,553,587,622,658,694,

%U 732,771,811,851,893,936,980,1024,1070,1117,1165,1213,1263,1314,1366,1418

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

%C Conjecture: the 2nd differences are eventually periodic: 1, 1, 1, 0, 2, 1, 0, 1, [2, 1, 1, 0].

%H Vaclav Kotesovec, <a href="/A293239/b293239.txt">Table of n, a(n) for n = 0..500</a>

%F Conjecture: a(n) ~ n^2/2. - _Vaclav Kotesovec_, Oct 05 2017

%F Conjectures from _Colin Barker_, Oct 05 2017: (Start)

%F G.f.: (1 + x^2 + x^3 + x^6 - x^8 + x^9 + x^12 - x^13) / ((1 - x)^2*(1 - x^4)).

%F 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 a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6) for n>6.

%F (End)

%e 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 Join[{1}, Length /@ Rest[NestList[Expand[D[#, x]] &, x^x, 53]]]

%Y Cf. A281434.

%K nonn

%O 0,2

%A _Vladimir Reshetnikov_, Oct 03 2017

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Last modified August 14 04:19 EDT 2024. Contains 375146 sequences. (Running on oeis4.)