A117209 -id:A117209 - cp's OEIS frontend

A320782 Inverse Euler transform of the unsigned Moebius function A008966.

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}]]

A306327 Expansion of Product_{k>=1} 1/(1 - mu(k)*x^k), where mu() is the Möbius function (A008683).

Original entry on oeis.org

1, 1, 0, -1, 0, 0, 1, 0, 0, -1, 2, 0, 1, -2, 2, 0, 4, -4, 1, -4, 6, -2, 8, -8, 6, -7, 13, -10, 13, -16, 17, -17, 22, -25, 29, -26, 40, -37, 40, -50, 58, -56, 69, -75, 82, -90, 108, -110, 128, -133, 158, -168, 185, -207, 229, -238, 281, -298, 328, -357, 405, -417, 477, -518, 564, -608

Offset: 0

Ilya Gutkovskiy, Feb 07 2019

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Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A008683, A073576, A117209.

nmax = 65; CoefficientList[Series[Product[1/(1 - MoebiusMu[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 65; CoefficientList[Series[Exp[Sum[Sum[MoebiusMu[j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d MoebiusMu[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 65}]

G.f.: exp(Sum_{k>=1} Sum_{j>=1} mu(j)^k*x^(j*k)/k).

A300673 Expansion of e.g.f. exp(Sum_{k>=1} mu(k)*x^k/k!), where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 1, 0, -3, -6, 5, 61, 126, -308, -2772, -5669, 25630, 224730, 486551, -3068155, -29264219, -72173176, 513535711, 5625869262, 16687752839, -113740116822, -1496118902963, -5508392724427, 31534346503605, 523333047780288, 2414704077547660, -10254467367668159

Offset: 0

Ilya Gutkovskiy, Mar 11 2018

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Exponential transform of A008683.

			E.g.f.: A(x) = 1 + x/1! - 3*x^3/3! - 6*x^4/4! + 5*x^5/5! + 61*x^6/6! + 126*x^7/7! - 308*x^8/8! - 2772*x^9/9! - 5669*x^10/10! + ...

Seiichi Manyama, Table of n, a(n) for n = 0..592
N. J. A. Sloane, Transforms

Cf. A008683, A050385, A050386, A068341, A073776, A104688, A117209, A117210, A195589, A204290, A300663.

nmax = 26; CoefficientList[Series[Exp[Sum[MoebiusMu[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[MoebiusMu[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 26}]

a(n) = if(n==0, 1, sum(k=1, n, moebius(k)*binomial(n-1, k-1)*a(n-k))); \\ Seiichi Manyama, Feb 27 2022

E.g.f.: exp(Sum_{k>=1} A008683(k)*x^k/k!).

a(0) = 1; a(n) = Sum_{k=1..n} mu(k) * binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Feb 27 2022

A307648 G.f. A(x) satisfies: 1/(1 - x) = A(x)A(x^2)^2A(x^3)^3A(x^4)^4 ... A(x^k)^k ...

Original entry on oeis.org

1, 1, -1, -4, -3, -2, 7, 7, 4, -6, 14, -11, -4, -47, 9, 6, 161, -93, -33, -269, 232, -83, 660, -733, 500, -779, 1527, -2291, 1876, -3892, 5598, -3056, 7791, -14088, 11289, -17113, 28083, -26211, 34645, -60715, 73180, -80951, 111926, -155269, 178561, -233709, 359679, -403884, 454659, -697310, 862133

Offset: 0

Ilya Gutkovskiy, Apr 19 2019

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Euler transform of A055615.

			G.f.: A(x) = 1 + x - x^2 - 4*x^3 - 3*x^4 - 2*x^5 + 7*x^6 + 7*x^7 + 4*x^8 - 6*x^9 + 14*x^10 - 11*x^11 - 4*x^12 - 47*x^13 + ...

Cf. A008683, A046970, A055615, A117209, A307649.

terms = 50; CoefficientList[Series[Product[1/(1 - x^k)^(MoebiusMu[k] k), {k, 1, terms}], {x, 0, terms}], x]
terms = 50; CoefficientList[Series[Exp[Sum[Sum[MoebiusMu[d] d^2, {d, Divisors[k]}] x^k/k, {k, 1, terms}]], {x, 0, terms}], x]
terms = 50; A[] = 1; Do[A[x] = 1/((1 - x) Product[A[x^k]^k, {k, 2, terms}]) + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]

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

G.f.: exp(Sum_{k>=1} A046970(k)*x^k/k).

A268649 G.f. A(x) satisfies: 1/(1-x) = Product_{n>=1} A( x^n/(1+x)^n ).

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 17, 36, 71, 143, 284, 573, 1140, 2287, 4568, 9138, 18272, 36559, 73098, 146216, 292413, 584836, 1169657, 2339353, 4678655, 9357356, 18714673, 37429377, 74858706, 149717506, 299434883, 598869895, 1197739689, 2395479446, 4790958784, 9581917760, 19163835261, 38327670814, 76655341388, 153310682944, 306621365618, 613242731721, 1226485462828, 2452970926285, 4905941852039, 9811883704440, 19623767408346, 39247534817726, 78495069634129

Offset: 0

Paul D. Hanna, Mar 26 2016

nonn

Compare g.f. to the identity: x = Sum_{n>=1} x^n/(1+x)^n.

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 9*x^5 + 17*x^6 + 36*x^7 + 71*x^8 + 143*x^9 + 284*x^10 + 573*x^11 + 1140*x^12 +...
where
1/(1-x) = A(x/(1+x)) * A(x^2/(1+x)^2) * A(x^3/(1+x)^3) * A(x^4/(1+x)^4) * A(x^5/(1+x)^5) *...
RELATED SERIES.
A(x/(1+x)) = 1 + x + x^3 + 2*x^5 - 4*x^6 + 14*x^7 - 35*x^8 + 86*x^9 - 191*x^10 +...
A(x^2/(1+x)^2) = 1 + x^2 - 2*x^3 + 4*x^4 - 8*x^5 + 17*x^6 - 38*x^7 + 88*x^8 +...
A(x^3/(1+x)^3) = 1 + x^3 - 3*x^4 + 6*x^5 - 9*x^6 + 9*x^7 - 26*x^9 + 72*x^10 +...
A(x^4/(1+x)^4) = 1 + x^4 - 4*x^5 + 10*x^6 - 20*x^7 + 36*x^8 - 64*x^9 + 120*x^10 +...
A(x^5/(1+x)^5) = 1 + x^5 - 5*x^6 + 15*x^7 - 35*x^8 + 70*x^9 - 125*x^10 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A117209.

{a(n) = my(A=[1,1],X=x+x*O(x^n)); for(i=1,n, A=concat(A,0); A[#A] = 1 - Vec( prod(k=1,#A, subst(Ser(A),x,x^k/(1+X)^k)) )[#A] );A[n+1]}
for(n=0,40,print1(a(n),", "))

G.f. satisfies: (1-x)/(1-2*x) = Product_{n>=1} A(x^n).

a(n) ~ c * 2^n, where c = 0.2788705076091492504414859194394933690344541628... . - Vaclav Kotesovec, Apr 02 2016

A307658 G.f. A(x) satisfies: (1 + x)/(1 - x) = A(x)A(x^2)A(x^3)A(x^4) ... A(x^k) ...

Original entry on oeis.org

1, 2, 0, -4, -4, 0, 4, 4, 0, -4, 0, 4, 0, -8, -4, 8, 16, 0, -20, -20, 8, 24, 20, -12, -24, -8, 24, 4, -16, -24, 16, 28, 24, -40, -32, 0, 72, 24, -28, -104, 0, 48, 88, -44, -32, -64, 92, 20, 24, -124, 64, 0, 96, -168, -12, -72, 272, -24, 72, -300, 104, -88, 316, -272, 128, -272, 376, -300

Offset: 0

Ilya Gutkovskiy, Apr 20 2019

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Convolution of A117209 and A117210.

			G.f.: A(x) = 1 + 2*x - 4*x^3 - 4*x^4 + 4*x^6 + 4*x^7 - 4*x^9 + 4*x^11 - 8*x^13 - 4*x^14 + 8*x^15 + ...

Cf. A008683, A117209, A117210, A307659.

terms = 67; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^MoebiusMu[k], {k, 1, terms}], {x, 0, terms}], x]
terms = 67; A[] = 1; Do[A[x] = (1 + x)/((1 - x) Product[A[x^k], {k, 2, terms}]) + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]

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

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}]]

A351402 G.f. A(x) satisfies: 1 / (1 - x) = Product_{i>=1, j>=1} A(x^(i*j)).

Original entry on oeis.org

1, 1, -1, -3, -1, 1, 4, 2, -2, -5, 4, 2, -2, -10, 3, 10, 21, -15, -26, -23, 34, 28, 25, -54, -18, 2, 67, -48, -22, -55, 116, 44, 37, -227, -10, 32, 295, -85, -76, -336, 254, 74, 250, -451, 59, -127, 672, -294, -69, -761, 740, 77, 657, -1208, 59, -450, 1700, -487, 241, -1892, 1202

Offset: 0

Ilya Gutkovskiy, Feb 10 2022

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Euler transform of A007427.

Cf. A000005, A006171, A007427, A101035, A117209, A351403.

nmax = 60; A007427[n_] := Sum[MoebiusMu[d] MoebiusMu[n/d], {d, Divisors[n]}]; CoefficientList[Series[Product[1/(1 - x^k)^A007427[k], {k, 1, nmax}], {x, 0, nmax}], x]

G.f. A(x) satisfies: 1 / (1 - x) = Product_{k>=1} A(x^k)^A000005(k).
G.f.: Product_{k>=1} 1 / (1 - x^k)^A007427(k).
G.f.: exp( Sum_{k>=1} A101035(k) * x^k / k ).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} A101035(k) * a(n-k).

A374364 Expansion of e.g.f. exp( x - Sum_{k>=1} x^(2^k)/2^k ).

Original entry on oeis.org

1, 1, 0, -2, -8, -24, 16, 400, -3072, -38528, -18944, 1287936, 17843200, 149045248, -188786688, -12007184384, -1265929355264, -20275964313600, 3871935889408, 2355175169523712, 45658709327609856, 565591105847689216, -1448855443865600000

Offset: 0

Seiichi Manyama, Jul 06 2024

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Seiichi Manyama, Table of n, a(n) for n = 0..200

Cf. A005388, A008683, A117209, A293604.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x-sum(k=1, ceil(log(N+1)/log(2)), x^2^k/2^k))))

E.g.f.: Product_{k>=1} (1 + x^(2*k-1))^(mu(2*k-1)/(2*k-1)), where mu() is the Moebius function.

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

sign

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]]

cp's OEIS Frontend

A320782 Inverse Euler transform of the unsigned Moebius function A008966.

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A306327 Expansion of Product_{k>=1} 1/(1 - mu(k)*x^k), where mu() is the Möbius function (A008683).

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A300673 Expansion of e.g.f. exp(Sum_{k>=1} mu(k)*x^k/k!), where mu() is the Moebius function (A008683).

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A307648 G.f. A(x) satisfies: 1/(1 - x) = A(x)*A(x^2)^2*A(x^3)^3*A(x^4)^4* ... *A(x^k)^k* ...

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A268649 G.f. A(x) satisfies: 1/(1-x) = Product_{n>=1} A( x^n/(1+x)^n ).

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A307658 G.f. A(x) satisfies: (1 + x)/(1 - x) = A(x)*A(x^2)*A(x^3)*A(x^4)* ... *A(x^k)* ...

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

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A351402 G.f. A(x) satisfies: 1 / (1 - x) = Product_{i>=1, j>=1} A(x^(i*j)).

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A307648 G.f. A(x) satisfies: 1/(1 - x) = A(x)A(x^2)^2A(x^3)^3A(x^4)^4 ... A(x^k)^k ...

A307658 G.f. A(x) satisfies: (1 + x)/(1 - x) = A(x)A(x^2)A(x^3)A(x^4) ... A(x^k) ...