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A322913 Inverse Moebius transform of the sequence (n*A032173(n+2): n >= 1).

%I A322913 #30 Apr 27 2020 14:44:44
%S A322913 1,3,7,15,36,81,197,455,1105,2618,6315,15141,36570,88161,213342,
%T A322913 516247,1251728,3037059,7378290,17938430,43655465,106317863,259127707,
%U A322913 631986437,1542364386,3766351332,9202390342,22496047757,55020807236,134631987776,329579227722,807142635031
%N A322913 Inverse Moebius transform of the sequence (n*A032173(n+2): n >= 1).
%C A322913 The sequence (A032173(n): n >= 1) shifts two places to the left under Bower's "CHK" (necklace, identity, unlabeled) transform. The current sequence satisfies A032173(n+2) = (1/n)*Sum_{d|n} mu(d)*a(n/d).
%H A322913 Andrew Howroyd, <a href="/A322913/b322913.txt">Table of n, a(n) for n = 1..200</a>
%H A322913 C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%F A322913 a(n) = Sum_{d|n} d*A032173(d+2).
%F A322913 a(n) = n*A032173(n) + Sum_{s=1..n-1} a(s)*A032173(n-s).
%F A322913 G.f.: If A(x) = Sum_{n>=1} a(n)*x^n and B(x) = Sum_{n>=1} A032173(n)*x^n, then A(x) = x*(dB(x)/dx)/(1-B(x)), while (B(x) - x - x^2)/x^2 = Sum_{n>=1} A032173(n+2)*x^n = -Sum_{n>=1} (mu(n)/n)*log(1-B(x^n)).
%t A322913 (* b = A032173 *) b[1] = b[2] = 1; c[1] = 1; c[2] = 3;
%t A322913 b[n_] := b[n] = 1/(n-2) Sum[MoebiusMu[(n-2)/d] c[d], {d, Divisors[n-2]}];
%t A322913 c[n_] := c[n] = n b[n] + Sum[c[s] b[n-s], {s, 1, n-1}];
%t A322913 a[n_] := Sum[d b[d+2], {d, Divisors[n]}];
%t A322913 Array[a, 26] (* _Jean-François Alcover_, Jan 02 2019 *)
%o A322913 (PARI)
%o A322913 CHK(p, n)={sum(d=1, n, moebius(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))}
%o A322913 seq(n)={my(p=1+O(x)); for(i=1, n\2, p=1+x+x*CHK(x*p, 2*i)); Vec(deriv(x*p)/(1-x*p)+O(x^n))} \\ _Andrew Howroyd_, Apr 27 2020
%Y A322913 Cf. A032171, A032173, A032174.
%K A322913 nonn
%O A322913 1,2
%A A322913 _Petros Hadjicostas_, Dec 30 2018
%E A322913 Terms a(27) and beyond from _Andrew Howroyd_, Apr 27 2020