A061255 -id:A061255 - 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

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.

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}]]

A301875 Expansion of Product_{k>=1} 1/(1 - x^k)^A007434(k).

Original entry on oeis.org

1, 1, 4, 12, 30, 78, 184, 448, 1033, 2361, 5292, 11676, 25382, 54470, 115508, 242132, 502520, 1032632, 2103172, 4246948, 8507968, 16915536, 33391788, 65470332, 127539321, 246928233, 475274592, 909658536, 1731703788, 3279644604, 6180528236

Offset: 0

Vaclav Kotesovec, Mar 28 2018

nonn

Euler transform of A007434.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A007434, A061255, A301876.

nmax = 40; CoefficientList[Series[Exp[Sum[Sum[Sum[d^2 MoebiusMu[k/d], {d, Divisors @ k}] * x^(j*k) / j, {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 31 2018 *)

a(n) ~ exp(4*Pi*n^(3/4) / (3^(5/4) * (5*Zeta(3))^(1/4)) + Zeta(3) / (2*Pi^2)) / (2^(3/2) * (15*Zeta(3))^(1/8) * n^(5/8)).

A318975 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^phi(k), where phi is the Euler totient function A000010.

Original entry on oeis.org

1, 2, 4, 10, 20, 42, 80, 154, 288, 522, 940, 1658, 2892, 4970, 8456, 14218, 23696, 39122, 64044, 104042, 167732, 268602, 427248, 675482, 1061632, 1659298, 2579676, 3990418, 6142892, 9412906, 14360136, 21814698, 33004704, 49739426, 74677924, 111713658

Offset: 0

Vaclav Kotesovec, Sep 06 2018

nonn

Convolution of A299069 and A061255.

			a(n) ~ exp(3^(4/3) * (7*Zeta(3))^(1/3) * n^(2/3) / (2*Pi^(2/3)) - 1/6) * A^2 * (7*Zeta(3))^(1/9) / (sqrt(2) * 3^(7/18) * Pi^(8/9) * n^(11/18)), where A is the Glaisher-Kinkelin constant A074962.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A061255, A299069, A301554, A301555.

Cf. A318976, A318814, A318977.

nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^EulerPhi[k], {k, 1, nmax}], {x, 0, nmax}], x]

A226106 G.f.: exp( Sum_{n>=1} A068963(n)*x^n/n ) where A068963(n) = Sum_{d|n} phi(d^3).

Original entry on oeis.org

1, 1, 3, 9, 20, 52, 105, 253, 536, 1142, 2421, 4999, 10278, 20686, 41512, 81984, 161029, 312681, 603070, 1153284, 2189331, 4129537, 7733317, 14399693, 26644337, 49034811, 89741600, 163411148, 296074694, 533909026, 958416113, 1712893825, 3048468607, 5403248469, 9539609984

Offset: 0

Paul D. Hanna, May 26 2013

nonn

Here phi(n) = A000010(n) is the Euler totient function.

Euler transform of A002618. - Vaclav Kotesovec, Mar 30 2018

			G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 20*x^4 + 52*x^5 + 105*x^6 + 253*x^7 +...
where
log(A(x)) = x + 5*x^2/2 + 19*x^3/3 + 37*x^4/4 + 101*x^5/5 + 95*x^6/6 + 295*x^7/7 + 293*x^8/8 + 505*x^9/9 +...+ A068963(n)*x^n/n +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, Graph - The asymptotic ratio

Cf. A068963, A002618, A061255, A299069, A301986.

nmax = 40; CoefficientList[Series[Product[1/(1-x^k)^(k*EulerPhi[k]), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 30 2018 *)
nmax = 40; CoefficientList[Series[Product[1/(1-x^k)^EulerPhi[k^2], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 30 2018 *)
nmax = 40; CoefficientList[Series[Exp[Sum[Sum[k*EulerPhi[k] * x^(j*k) / j, {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 31 2018 *)

{a(n)=polcoeff(exp(sum(m=1,n+1,sumdiv(m,d,eulerphi(d^3))*x^m/m)+x*O(x^n)),n)}
for(n=0,35,print1(a(n),", "))

a(n) ~ exp(2^(9/4) * sqrt(Pi) * n^(3/4) / (3 * 5^(1/4)) + 3*Zeta(3) / Pi^2) / (2^(11/8) * 5^(1/8) * Pi^(1/4) * n^(5/8)). - Vaclav Kotesovec, Mar 30 2018

A318811 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)*x^k), where phi is the Euler totient function A000010.

Original entry on oeis.org

1, 1, 3, 19, 121, 1161, 9931, 124363, 1542129, 21594961, 335083411, 5712781251, 104044684393, 2036445474649, 42781075481691, 943820382272251, 22433542236603361, 556276331238284193, 14612462927067954979, 401110580118493111411, 11553483337639043003481

Offset: 0

Vaclav Kotesovec, Sep 04 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..435
Eric Weisstein's World of Mathematics, Totient Function.
Wikipedia, Euler's totient function.

Cf. A061255, A061256, A006171.
Cf. A299069, A192065, A107742.
Cf. A294361, A294363.

nmax = 25; CoefficientList[Series[Exp[Sum[EulerPhi[k]*x^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]!

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, eulerphi(k)*x^k)))) \\ Seiichi Manyama, Apr 07 2022

a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, k*eulerphi(k)*a(n-k)/(n-k)!)); \\ Seiichi Manyama, Apr 07 2022

a(n) ~ 2^(1/3) * exp(1/6 + 3^(4/3) * n^(2/3) / (2^(1/3) * Pi^(2/3)) - n) * n^(n - 1/6) / (3*Pi)^(1/3).

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} k * phi(k) * a(n-k)/(n-k)!. - Seiichi Manyama, Apr 07 2022

A381679 Euler transform of A000056.

Original entry on oeis.org

1, 1, 7, 31, 100, 364, 1152, 3864, 12102, 37358, 113618, 337562, 990798, 2857926, 8144334, 22902470, 63660695, 175026047, 476242001, 1283435153, 3427047146, 9072455146, 23820491998, 62057045134, 160471504373, 412022656517, 1050740365571, 2662223436203

Offset: 0

Seiichi Manyama, Mar 04 2025

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A061255, A381680.

Cf. A000056, A084218, A156733.

a[0] = 1; a[n_] := a[n] = Sum[DivisorSigma[4, k^2]/DivisorSigma[2, k^2]*a[n-k], {k, 1, n}]/n; Table[a[n], {n, 0, 30}] (* Vaclav Kotesovec, Mar 04 2025 *)

my(N=30, x='x+O('x^N)); Vec(exp(sum(k=1, N, sigma(k^2, 4)/sigma(k^2, 2)*x^k/k)))

G.f.: 1/Product_{k>=1} (1 - x^k)^A000056(k).
G.f.: exp( Sum_{k>=1} sigma_4(k^2)/sigma_2(k^2) * x^k/k ).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} sigma_4(k^2)/sigma_2(k^2) * a(n-k).
a(n) ~ exp(5*(3*zeta(5)/zeta(3))^(1/5) * n^(4/5) / 2^(7/5) - 1/10 - 12*zeta'(-3)) * A^(6/5) * (3*zeta(5)/zeta(3))^(3/25) / (2^(7/50) * sqrt(5*Pi) * n^(31/50)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Mar 04 2025

A301986 Expansion of Product_{k>=1} (1 + x^k)^(k*A000010(k)), where A000010 is the Euler totient function.

Original entry on oeis.org

1, 1, 2, 8, 15, 41, 75, 179, 378, 748, 1591, 3133, 6369, 12357, 24225, 46691, 89301, 169589, 318413, 596255, 1103468, 2036880, 3725353, 6786021, 12281026, 22107132, 39604155, 70566697, 125209095, 221048851, 388705826, 680465440, 1186649341, 2061086935

Offset: 0

Vaclav Kotesovec, Mar 30 2018

nonn

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

Cf. A002618, A061255, A226106, A299069.

nmax = 40; CoefficientList[Series[Product[(1+x^k)^(k*EulerPhi[k]), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^(j + 1)/j * Sum[k*EulerPhi[k] * x^(j*k), {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x]

a(n) ~ exp(2^(3/2) * 7^(1/4) * sqrt(Pi) * n^(3/4) / (3 * 5^(1/4))) * 7^(1/8) / (2^(7/4) * 5^(1/8) * Pi^(1/4) * n^(5/8)).

A319111 Expansion of Product_{k>=1} 1/(1 - phi(k)*x^k), where phi = Euler totient function (A000010).

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 22, 38, 63, 105, 174, 278, 447, 707, 1122, 1766, 2729, 4213, 6482, 9880, 15069, 22799, 34290, 51378, 76777, 114365, 169324, 250162, 368505, 540575, 792042, 1154798, 1680385, 2439101, 3530308, 5103380, 7349875, 10564955, 15155752, 21696072, 31007949, 44199845

Offset: 0

Ilya Gutkovskiy, Sep 10 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..8200

Cf. A000010, A061255, A159929, A318917.

Cf. A279784, A306327, A316961.

with(numtheory): a:=series(mul(1/(1-phi(k)*x^k),k=1..50),x=0,42): seq(coeff(a,x,n),n=0..41); # Paolo P. Lava, Apr 02 2019

nmax = 41; CoefficientList[Series[Product[1/(1 - EulerPhi[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 41; CoefficientList[Series[Exp[Sum[Sum[EulerPhi[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 EulerPhi[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 41}]

G.f.: exp(Sum_{k>=1} Sum_{j>=1} phi(j)^k*x^(j*k)/k).
From Vaclav Kotesovec, Feb 08 2019: (Start)
a(n) ~ c * 2^(2*n/5), where
c = 18827.6460615531202942792897255332975807324818737172163... if mod(n,5) = 0
c = 18827.5079339024144115146595255453426552477117955925738... if mod(n,5) = 1
c = 18827.4967567108036710998657106724179082561779712900405... if mod(n,5) = 2
c = 18827.4818413568083742650057347700058389606441225811016... if mod(n,5) = 3
c = 18827.4547665561882994953942505862213438332903500716893... if mod(n,5) = 4
(End)

A308457 Expansion of e.g.f. (1/(1 - x)) * Product_{k>=2} 1/(1 - x^k)^(phi(k)/2), where phi() is the Euler totient function (A000010).

Original entry on oeis.org

1, 1, 3, 15, 93, 765, 6615, 73395, 855225, 11348505, 163593675, 2633729175, 44537325525, 829112008725, 16299062754975, 340762189642875, 7597436750528625, 178862527106888625, 4426363064514265875, 115222810432347993375, 3139125774622690978125

Offset: 0

Ilya Gutkovskiy, May 27 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..435

Cf. A000010, A023022, A023896, A051193, A057661, A061255.

nmax = 20; CoefficientList[Series[1/(1 - x) Product[1/(1 - x^k)^(EulerPhi[k]/2), {k, 2, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[Exp[Sum[Sum[LCM[k, j], {j, 1, k}] x^k/k^2, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Total[Numerator[Range[k]/k]] k! Binomial[n - 1, k - 1] a[n - k]/k, {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 20}]

E.g.f.: exp(Sum_{k>=1} A057661(k)*x^k/k).
E.g.f.: exp(Sum_{k>=1} A051193(k)*x^k/k^2).
E.g.f.: d/dx ( exp(arctanh(x)) ) * Product_{k>=3} 1/(1 - x^k)^A023022(k).
a(n) ~ A * exp(3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / (2*Pi)^(2/3) - n - 1/12) * n^(n + 1/36) / (2^(1/9) * 3^(19/36) * (Pi*Zeta(3))^(1/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, May 28 2019
E.g.f.: Product_{k>=1} 1/(1 - x^k)^(A023896(k)/k). - Ilya Gutkovskiy, May 28 2019

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}]]

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A320782 Inverse Euler transform of the unsigned Moebius function A008966.

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A301875 Expansion of Product_{k>=1} 1/(1 - x^k)^A007434(k).

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A318975 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^phi(k), where phi is the Euler totient function A000010.

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A226106 G.f.: exp( Sum_{n>=1} A068963(n)*x^n/n ) where A068963(n) = Sum_{d|n} phi(d^3).

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A318811 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)*x^k), where phi is the Euler totient function A000010.

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A381679 Euler transform of A000056.

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A301986 Expansion of Product_{k>=1} (1 + x^k)^(k*A000010(k)), where A000010 is the Euler totient function.

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A319111 Expansion of Product_{k>=1} 1/(1 - phi(k)*x^k), where phi = Euler totient function (A000010).

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