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A032173 Sequence (a(n): n >= 1) that shifts left 2 places under the "CHK" (necklace, identity, unlabeled) transform and has initial terms a(1) = a(2) = 1.

%I A032173 #37 Jan 02 2019 11:53:25
%S A032173 1,1,1,1,2,3,7,12,28,55,122,258,574,1254,2813,6283,14220,32237,73631,
%T A032173 168660,388331,896790,2078822,4832343,11266422,26332119,61694574,
%U A032173 144858260,340829231,803427128,1897269215,4487725726
%N A032173 Sequence (a(n): n >= 1) that shifts left 2 places under the "CHK" (necklace, identity, unlabeled) transform and has initial terms a(1) = a(2) = 1.
%C A032173 From _Petros Hadjicostas_, Dec 29 2018: (Start)
%C A032173 a(n+2) = (1/n)*Sum_{d|n} mu(n/d)*c(d), where c(n) = n*a(n) + Sum_{s=1..n-1} c(s)*a(n-s) with a(1) = a(2) = 1, c(1) = 1, and c(2) = 3.
%C A032173 G.f.: If A(x) = Sum_{n>=1} a(n)*x^n, then Sum_{n>=1} a(n+2)*x^n = -Sum_{n>=1} (mu(n)/n)*log(1-A(x^n)).
%C A032173 The g.f. of the auxiliary sequence (c(n): n>=1) is C(x) = Sum_{n>=1} c(n)*x^n = x*(dA(x)/dx)/(1-A(x)) = x + 3*x^2 + 7*x^3 + 15*x^4 + 36*x^5 + 81*x^6 + 197*x^7 + 455*x^8 + 1105*x^9 + 2618*x^10 + ... (The auxiliary sequence is given by sequence A322913.)
%C A032173 (End)
%C A032173 The first two terms of the sequence must be specified. In general, if the sequence (b(n): n >= 1) is such that (b(n+2): n >= 1) = CHK((b(n): n >= 1)), then b(3) = b(1), b(4) = (1/2)*(b(1)^2 + 2*b(2) - b(1)), b(5) = (b(1)/3)*(b(1)^2 + 3*b(2) + 2), and so on. - _Petros Hadjicostas_, Dec 31 2018
%H A032173 Andrew Howroyd, <a href="/A032173/b032173.txt">Table of n, a(n) for n = 1..200</a>
%H A032173 C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%t A032173 a[1] = a[2] = 1; c[1] = 1; c[2] = 3;
%t A032173 a[n_] := a[n] = 1/(n-2) Sum[MoebiusMu[(n-2)/d] c[d], {d, Divisors[n-2]}];
%t A032173 c[n_] := c[n] = n a[n] + Sum[c[s] a[n-s], {s, 1, n-1}];
%t A032173 Array[a, 32] (* _Jean-François Alcover_, Jan 02 2019 *)
%o A032173 (PARI)
%o A032173 CHK(p,n)={sum(d=1, n, moebius(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))}
%o A032173 seq(n)={my(p=1+O(x));for(i=1, n\2, p=1+x+x*CHK(x*p, 2*i)); Vec(p+O(x^n))} \\ _Andrew Howroyd_, Jun 20 2018
%Y A032173 Cf. A032171, A032174, A322913.
%K A032173 nonn
%O A032173 1,5
%A A032173 _Christian G. Bower_
%E A032173 Name modified by _Petros Hadjicostas_, Jan 01 2019