A293548 -id:A293548 - cp's OEIS frontend

A320785 Inverse Euler transform of the number of factorizations function A001055.

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

Offset: 0

Gus Wiseman, Oct 22 2018

sign

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.

OEIS Wiki, Euler transform

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, A320778, A320779, A320780, A320781, A320782.

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}]]

A320786 Inverse Euler transform of {1,0,1,0,0,0,...}.

Original entry on oeis.org

1, 1, -1, 1, -1, 1, -2, 2, -2, 3, -5, 6, -7, 11, -16, 20, -27, 39, -55, 75, -102, 145, -207, 286, -397, 565, -802, 1123, -1581, 2248, -3193, 4517, -6399, 9112, -12984, 18457, -26270, 37502, -53553, 76416, -109146, 156135, -223446, 319764, -457884, 656288, -941081

Offset: 0

Gus Wiseman, Oct 22 2018

sign

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.

OEIS Wiki, Euler transform

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, A320778, A320779, A320780, A320781, A320782.

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[PadRight[{1,0,1},50]]

A327725 Expansion of Product_{i>=1, j>=1} (1 + x^(i*prime(j))).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 3, 3, 4, 6, 7, 9, 12, 14, 18, 24, 27, 35, 43, 51, 64, 79, 92, 113, 137, 162, 195, 235, 276, 329, 394, 460, 546, 646, 753, 890, 1044, 1214, 1422, 1662, 1927, 2245, 2611, 3015, 3497, 4051, 4662, 5385, 6209, 7128, 8203, 9423, 10786

Offset: 0

Ilya Gutkovskiy, Sep 23 2019

nonn

Weigh transform of A001221.

Cf. A001221, A293548.

nmax = 52; CoefficientList[Series[Product[(1 + x^k)^PrimeNu[k], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d PrimeNu[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 52}]

seq(n)={Vec(prod(k=1, n, (1 + x^k + O(x*x^n))^omega(k)))} \\ Andrew Howroyd, Sep 23 2019

G.f.: Product_{k>=1} (1 + x^k)^A001221(k).

cp's OEIS Frontend

A320785 Inverse Euler transform of the number of factorizations function A001055.

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A320786 Inverse Euler transform of {1,0,1,0,0,0,...}.

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A327725 Expansion of Product_{i>=1, j>=1} (1 + x^(i*prime(j))).

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