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A320785
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Inverse Euler transform of the number of factorizations function A001055.
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0
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1, 1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 1, -1, 0, 0, 1, 1, -3, 3, -3, 0, 4, -6, 6, -5, 5, -1, -7, 13, -16, 15, -8, -3, 12, -25, 41, -40, 21, 10, -51, 83, -93, 81, -38, -44, 148, -234, 258, -190, 35, 184, -429, 616, -660, 480, -18, -640, 1289, -1714, 1693, -1039, -268
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OFFSET
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0,18
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COMMENTS
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The Euler transform of a sequence q is the sequence of coefficients of x^n, n > 0, in the expansion of Product_{n > 0} 1/(1 - x^n)^q(n). The constant term 1 is sometimes taken to be the zeroth part of the Euler transform.
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LINKS
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MATHEMATICA
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EulerInvTransform[{}]={}; EulerInvTransform[seq_]:=Module[{final={}}, For[i=1, i<=Length[seq], i++, AppendTo[final, i*seq[[i]]-Sum[final[[d]]*seq[[i-d]], {d, i-1}]]];
Table[Sum[MoebiusMu[i/d]*final[[d]], {d, Divisors[i]}]/i, {i, Length[seq]}]];
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
EulerInvTransform[Table[Length[facs[n]], {n, 100}]]
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CROSSREFS
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Euler transforms: A000081, A001970, A006171, A007294, A061255, A061256, A061257, A073576, A117209, A293548, A293549.
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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