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A320778 Inverse Euler transform of the Euler totient function phi = A000010. 8
1, 1, 0, 1, 0, 2, -3, 4, -4, 4, -9, 14, -19, 30, -42, 50, -76, 128, -194, 286, -412, 598, -909, 1386, -2100, 3178, -4763, 7122, -10758, 16414, -25061, 38056, -57643, 87568, -133436, 203618, -311128, 475536, -726355, 1109718, -1697766, 2601166, -3987903, 6114666 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..43.

MATHEMATICA

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]}]];

EulerInvTransform[Array[EulerPhi, 30]]

CROSSREFS

Number theoretical functions: A000005, A000010, A000203, A001055, A001221, A001222, A008683, A010054.

Euler transforms: A000081, A001970, A006171, A007294, A061255, A061256, A061257, A073576, A117209, A293548, A293549.

Inverse Euler transforms: A059966, A320767, A320776, A320777, A320779, A320780, A320781, A320782.

Sequence in context: A309941 A211509 A305594 * A058277 A065852 A303998

Adjacent sequences:  A320775 A320776 A320777 * A320779 A320780 A320781

KEYWORD

sign

AUTHOR

Gus Wiseman, Oct 22 2018

STATUS

approved

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Last modified February 20 19:02 EST 2020. Contains 332082 sequences. (Running on oeis4.)