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A060224 Number of orbits of length n under the map whose periodic points are counted by A047863.

%I A060224 #17 Jan 05 2025 19:51:36
%S A060224 2,2,8,39,288,3046,47232,1061100,34385064,1601137110,106806380544,
%T A060224 10186152828755,1386394018652160,268976332493883474,
%U A060224 74301040560350828856,29201332000320392849280,16315436194909017151242240,12952804290011844088808158188,14603450579455204338154338779136
%N A060224 Number of orbits of length n under the map whose periodic points are counted by A047863.
%H A060224 G. C. Greubel, <a href="/A060224/b060224.txt">Table of n, a(n) for n = 1..113</a>
%H A060224 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H A060224 Yash Puri and Thomas Ward, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/39-5/puri.pdf">A dynamical property unique to the Lucas sequence</a>, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.
%H A060224 T. Ward, <a href="http://www.mth.uea.ac.uk/~h720/research/files/integersequences.html">Exactly realizable sequences</a>
%F A060224 a(n) = (1/n)* Sum_{ d divides n } mu(d)*A047863(n/d).
%e A060224 a(5)=288 since the 6th term of A047863 is 1442 and the 2nd term is 2, so there must be (1442-2)/5 = 288 orbits of length 5.
%t A060224 A047863[n_]:= A047863[n]= Sum[Binomial[n,k]*2^(k*(n-k)), {k,0,n}];
%t A060224 A060224[n_]:= DivisorSum[n, MoebiusMu[#]*A047863[n/#] &]/n;
%t A060224 Table[A060224[n], {n,40}] (* _G. C. Greubel_, Nov 03 2024 *)
%o A060224 (PARI) a047863(n) = n!*polcoeff(sum(k=0, n, exp(2^k*x +x*O(x^n))*x^k/k!), n);
%o A060224 a(n) = (1/n)*sumdiv(n, d, moebius(d)*a047863(n/d)); \\ _Michel Marcus_, Sep 11 2017
%o A060224 (Magma)
%o A060224 A047863:= func< n | (&+[Binomial(n,k)*2^(k*(n-k)): k in [0..n]]) >;
%o A060224 A060224:= func< n | (&+[MoebiusMu(d)*A047863(Floor(n/d)): d in Divisors(n)])/n >;
%o A060224 [A060224(n): n in [1..40]]; // _G. C. Greubel_, Nov 03 2024
%o A060224 (SageMath)
%o A060224 def A047863(n): return sum(binomial(n,k)*2^(k*(n-k)) for k in range(n+1))
%o A060224 def A060224(n): return sum(moebius(k)*A047863(n//k) for k in (1..n) if (k).divides(n))//n
%o A060224 [A060224(n) for n in range(1,41)] # _G. C. Greubel_, Nov 03 2024
%Y A060224 Cf. A008683, A047863.
%K A060224 nonn
%O A060224 1,1
%A A060224 _Thomas Ward_, Mar 21 2001
%E A060224 More terms from _Michel Marcus_, Sep 11 2017