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A322913
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Inverse Moebius transform of the sequence (n*A032173(n+2): n >= 1).
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2
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1, 3, 7, 15, 36, 81, 197, 455, 1105, 2618, 6315, 15141, 36570, 88161, 213342, 516247, 1251728, 3037059, 7378290, 17938430, 43655465, 106317863, 259127707, 631986437, 1542364386, 3766351332, 9202390342, 22496047757, 55020807236, 134631987776, 329579227722, 807142635031
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OFFSET
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1,2
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COMMENTS
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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).
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LINKS
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FORMULA
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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)).
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MATHEMATICA
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(* b = A032173 *) b[1] = b[2] = 1; c[1] = 1; c[2] = 3;
b[n_] := b[n] = 1/(n-2) Sum[MoebiusMu[(n-2)/d] c[d], {d, Divisors[n-2]}];
c[n_] := c[n] = n b[n] + Sum[c[s] b[n-s], {s, 1, n-1}];
a[n_] := Sum[d b[d+2], {d, Divisors[n]}];
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PROG
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(PARI)
CHK(p, n)={sum(d=1, n, moebius(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))}
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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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