A320767 -id:A320767 - cp's OEIS frontend

A292518 Expansion of Product_{k>=1} (1 - x^(k*(k+1)/2)).

1, -1, 0, -1, 1, 0, -1, 1, 0, 1, -2, 1, 0, 1, -1, -1, 2, -1, 1, -2, 1, 0, 0, 0, 0, 1, -1, 1, -3, 2, -1, 2, -1, 0, 1, -1, 0, -2, 3, -1, 1, -2, 1, 1, -2, 0, 0, 2, 0, -1, 0, 2, -2, -1, -1, 1, 2, -1, 1, -1, 1, -2, 1, -2, 3, 1, -2, 0, -2, 3, -1, -1, 0, 3, -1, 0, -2, 1, 0, -3, 2, 2, 1, -1, -1, 0, 0, -1, 0, 2, -1

Offset: 0

Ilya Gutkovskiy, Sep 18 2017

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Convolution inverse of A007294.
The difference between the number of partitions of n into an even number of distinct triangular numbers and the number of partitions of n into an odd number of distinct triangular numbers.
Euler transform of {-1 if n is a triangular number else 0, n > 0} = -A010054. - Gus Wiseman, Oct 22 2018

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for related partition-counting sequences

Product_{k>=1} (1 - x^(k*((m-2)*k-(m-4))/2)): this sequence (m=3), A276516 (m=4), A305355 (m=5).

Cf. A007294, A010054, A024940, A280366, A320767, A320784.

nmax = 90; CoefficientList[Series[Product[1 - x^(k (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 - x^(k*(k+1)/2)).

A320778 Inverse Euler transform of the Euler totient function phi = A000010.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Oct 22 2018

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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.

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.

# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(n -> ifelse(n=0, 1, NumberTheory:-Totient(n))):
seq(a(n), n = 0..43); # Peter Luschny, Nov 21 2022

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]]

A320776 Inverse Euler transform of the number of prime factors (with multiplicity) function A001222.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, -1, -1, 0, 1, 0, -1, -1, -1, 1, 3, 3, -2, -5, -4, 0, 7, 7, 0, -9, -10, 2, 15, 15, -3, -27, -30, 3, 46, 51, 1, -71, -91, -7, 117, 157, 23, -194, -265, -57, 318, 465, 111, -536, -821, -230, 893, 1456, 505, -1485, -2559, -1036, 2433, 4483, 2022

Offset: 0

Gus Wiseman, Oct 22 2018

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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.

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, A320777, A320778, A320779, A320780, A320781, A320782.

# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(n -> ifelse(n=0, 1, NumberTheory:-NumberOfPrimeFactors(n))):
seq(a(n), n = 0..59); # Peter Luschny, Nov 21 2022

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[PrimeOmega,100]]

A320777 Inverse Euler transform of the number of distinct prime factors (without multiplicity) function A001221.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 0, 0, -1, -1, 1, 1, 0, -1, 0, 1, -1, -2, 1, 3, 1, -2, -2, 1, 0, -4, 0, 6, 6, -4, -8, 1, 4, -4, -5, 10, 16, -4, -25, -7, 17, 5, -16, 2, 42, 12, -58, -48, 40, 59, -27, -44, 67, 86, -103, -187, 36, 236, 45, -213, -5, 284, -23, -526, -188, 663, 520

Offset: 0

Gus Wiseman, Oct 22 2018

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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.

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, 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[Array[PrimeNu,100]]

A320782 Inverse Euler transform of the unsigned Moebius function A008966.

Original entry on oeis.org

1, 1, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, -2, 3, 0, -1, -3, 6, -3, 0, -6, 12, -6, 0, -9, 23, -17, 0, -15, 47, -40, 8, -24, 91, -101, 34, -46, 181, -230, 109, -92, 354, -534, 323, -208, 690, -1177, 883, -520, 1365, -2603, 2297, -1377, 2760, -5641, 5789, -3721, 5741

Offset: 0

Gus Wiseman, Oct 22 2018

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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.

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.

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[Table[Abs[MoebiusMu[n]],{n,30}]]

A320784 Negated inverse Euler transform of {-1 if n is a triangular number else 0, n > 0} = -A010054.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 2, 3, 3, 5, 8, 11, 14, 23, 31, 47, 68, 101, 144, 217, 315, 471, 693, 1035, 1528, 2287, 3397, 5085, 7587, 11377, 17017, 25565, 38349, 57681, 86724, 130645, 196778, 296853, 447864, 676479, 1022082, 1545685, 2338299, 3540111, 5361606, 8125551

Offset: 0

Gus Wiseman, Oct 22 2018

nonn

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[-Table[SquaresR[1,8*n+1]/2,{n,30}]]

A320783 Inverse Euler transform of (-1)^(n - 1).

Original entry on oeis.org

1, 1, -2, 2, -3, 6, -11, 18, -30, 56, -105, 186, -335, 630, -1179, 2182, -4080, 7710, -14588, 27594, -52377, 99858, -190743, 364722, -698870, 1342176, -2581425, 4971008, -9586395, 18512790, -35792449, 69273666, -134215680, 260300986, -505294125, 981706806

Offset: 0

Gus Wiseman, Oct 22 2018

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After a(1) and a(2), same as A038063.

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[Array[(-1)^(#-1)&,30]]

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

Original entry on oeis.org

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]]

cp's OEIS Frontend

A292518 Expansion of Product_{k>=1} (1 - x^(k*(k+1)/2)).

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A320778 Inverse Euler transform of the Euler totient function phi = A000010.

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A320776 Inverse Euler transform of the number of prime factors (with multiplicity) function A001222.

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Maple

Mathematica

A320777 Inverse Euler transform of the number of distinct prime factors (without multiplicity) function A001221.

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Mathematica

A320782 Inverse Euler transform of the unsigned Moebius function A008966.

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Mathematica

A320784 Negated inverse Euler transform of {-1 if n is a triangular number else 0, n > 0} = -A010054.

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A320783 Inverse Euler transform of (-1)^(n - 1).

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

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Mathematica

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

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