A038070 -id:A038070 - cp's OEIS frontend

A038063 Product_{k>=1}1/(1 - x^k)^a(k) = 1 + 2x.

2, -3, 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

Offset: 1

Christian G. Bower, Jan 04 1999

sign

Apart from initial terms, exponents in expansion of A065472 as a product zeta(n)^(-a(n)).

Seiichi Manyama, Table of n, a(n) for n = 1..3000
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
N. J. A. Sloane, Euler transform.

Cf. A001037, A008683, A038064, A038065, A038066, A038067, A038068, A038069, A038070, A059966, A060477, A065472.

a[n_] := DivisorSum[n, (-1)^(#+1) * MoebiusMu[n/#]*2^# &] / n; Array[a, 33] (* Amiram Eldar, May 29 2025 *)

{a(n)=polcoeff(sum(k=1,n,moebius(k)/k*log(1+2*x^k+x*O(x^n))),n)} \\ Paul D. Hanna, Oct 13 2010

a(n) = (1/n) * Sum_{d divides n} (-1)^(d+1)*moebius(n/d)*2^d. - Vladeta Jovovic, Sep 06 2002
G.f.: Sum_{n>=1} moebius(n)*log(1 + 2*x^n)/n, where moebius(n) = A008683(n). - Paul D. Hanna, Oct 13 2010
For n == 0, 1, 3 (mod 4), a(n) = (-1)^(n+1)*A001037(n), which for n>1 also equals (-1)^(n+1)*A059966(n) = (-1)^(n+1)*A060477(n).
For n == 2 (mod 4), a(n) = -(A001037(n) + A001037(n/2)). - George Beck and Max Alekseyev, May 23 2016
a(n) ~ -(-1)^n * 2^n / n. - Vaclav Kotesovec, Jun 12 2018

A038064 Product_{k>=1} 1/(1 - x^k)^a(k) = 1 + 3x.

Original entry on oeis.org

3, -6, 8, -18, 48, -124, 312, -810, 2184, -5928, 16104, -44220, 122640, -341796, 956576, -2690010, 7596480, -21524412, 61171656, -174336264, 498111952, -1426419852, 4093181688, -11767874940, 33891544368, -97764131640

Offset: 1

Christian G. Bower, Jan 04 1999

sign

Seiichi Manyama, Table of n, a(n) for n = 1..2000
N. J. A. Sloane, Euler transform

Cf. A038063, A038064, A038065, A038066, A038067, A038068, A038069, A038070.

{a(n)=polcoeff(sum(k=1,n,moebius(k)/k*log(1+3*x^k+x*O(x^n))),n)} \\ Paul D. Hanna, Oct 13 2010

G.f.: Sum_{n>=1} moebius(n)*log(1 + 3*x^n)/n, where moebius(n)=A008683(n). - Paul D. Hanna, Oct 13 2010

a(n) = -(1/n) * Sum_{d|n} mu(n/d) * (-3)^d. - Seiichi Manyama, Apr 12 2025

A038065 Product_{k>=1} 1/(1 - x^k)^a(k) = 1 + 4x.

Original entry on oeis.org

4, -10, 20, -60, 204, -690, 2340, -8160, 29120, -104958, 381300, -1397740, 5162220, -19175130, 71582716, -268431360, 1010580540, -3817763040, 14467258260, -54975528948, 209430785460, -799645010850, 3059510616420

Offset: 1

Christian G. Bower, Jan 04 1999

sign

Seiichi Manyama, Table of n, a(n) for n = 1..1000
N. J. A. Sloane, Euler transform

Cf. A038063, A038064, A038066, A038067, A038068, A038069, A038070.

x='x+O('x^24); Vec(sum(k=1, 24, moebius(k)*log(1 + 4*x^k)/k)) \\ Indranil Ghosh, May 24 2017

G.f.: Sum_{k>=1} mu(k)*log(1 + 4*x^k)/k. - Ilya Gutkovskiy, May 23 2017
a(n) ~ -(-1)^n * 4^n / n. - Vaclav Kotesovec, Jun 12 2018
a(n) = -(1/n) * Sum_{d|n} mu(n/d) * (-4)^d. - Seiichi Manyama, Apr 12 2025

A038066 Product_{k>=1} 1/(1 - x^k)^a(k) = 1 + 5x.

Original entry on oeis.org

5, -15, 40, -150, 624, -2620, 11160, -48750, 217000, -976872, 4438920, -20343700, 93900240, -435970980, 2034504992, -9536718750, 44878791360, -211927733500, 1003867701480, -4768371093720, 22706531339280

Offset: 1

Christian G. Bower, Jan 04 1999

sign

Seiichi Manyama, Table of n, a(n) for n = 1..1000
N. J. A. Sloane, Euler transform

Cf. A038063, A038064, A038065, A038067, A038068, A038069, A038070.

x='x+O('x^22); Vec(sum(k=1, 22, moebius(k)*log(1 + 5*x^k)/k)) \\ Indranil Ghosh, May 24 2017

G.f.: Sum_{k>=1} mu(k)*log(1 + 5*x^k)/k. - Ilya Gutkovskiy, May 23 2017
a(n) ~ -(-1)^n * 5^n / n. - Vaclav Kotesovec, Jun 12 2018
a(n) = -(1/n) * Sum_{d|n} mu(n/d) * (-5)^d. - Seiichi Manyama, Apr 12 2025

A038067 Product_{k>=1} (1 + x^k)^a(k) = 1 + 2x.

Original entry on oeis.org

2, -1, 2, -4, 6, -9, 18, -34, 56, -99, 186, -344, 630, -1161, 2182, -4114, 7710, -14532, 27594, -52476, 99858, -190557, 364722, -699214, 1342176, -2580795, 4971008, -9587556, 18512790, -35790267, 69273666, -134219794, 260300986

Offset: 1

Christian G. Bower, Jan 04 1999

sign

Seiichi Manyama, Table of n, a(n) for n = 1..3000
Laura Pierson, A power sum expansion for the Kromatic symmetric function, arXiv:2408.01395 [math.CO], 2024. See p. 5.
N. J. A. Sloane, Weigh transform

Cf. A038063-A038070, A306156.

Dirichlet convolution of A038063 with characteristic function of powers of 2.

a(n) = (1/n) * (-(-2)^n + Sum_{dVladeta Jovovic, Sep 06 2002

A038068 Product_{k>=1}(1 + x^k)^a(k) = 1 + 3x.

Original entry on oeis.org

3, -3, 8, -21, 48, -116, 312, -831, 2184, -5880, 16104, -44336, 122640, -341484, 956576, -2690841, 7596480, -21522228, 61171656, -174342144, 498111952, -1426403748, 4093181688, -11767919276, 33891544368, -97764009000

Offset: 1

Christian G. Bower, Jan 04 1999

sign

Seiichi Manyama, Table of n, a(n) for n = 1..2000
Christian G. Bower, PARI programs for transforms, 2007.
N. J. A. Sloane, Maple programs for transforms, 2001-2020.

Cf. A038063-A038070, A306157.

Dirichlet convolution of A038064 with characteristic function of powers of 2.

a(n) = (1/n) * (-(-3)^n + Sum_{dSeiichi Manyama, Jun 23 2018

A038069 Product_{k>=1} ((1 + x^k)^a(k)) = 1 + 4x.

Original entry on oeis.org

4, -6, 20, -66, 204, -670, 2340, -8226, 29120, -104754, 381300, -1398410, 5162220, -19172790, 71582716, -268439586, 1010580540, -3817733920, 14467258260, -54975633702, 209430785460, -799644629550, 3059510616420

Offset: 1

Christian G. Bower, Jan 04 1999

sign

Seiichi Manyama, Table of n, a(n) for n = 1..1000
N. J. A. Sloane, Weigh transform

Cf. A038063-A038070, A306158.

Dirichlet convolution of A038065 with characteristic function of powers of 2.

a(n) = (1/n)*(-(-4)^n + Sum_{dSeiichi Manyama, Jun 23 2018

A306159 Inverse Weigh transform of 5^n.

Original entry on oeis.org

5, 15, 40, 165, 624, 2620, 11160, 48915, 217000, 976872, 4438920, 20346320, 93900240, 435970980, 2034504992, 9536767665, 44878791360, 211927733500, 1003867701480, 4768372070592, 22706531339280, 108372083629260, 518301258916440, 2483526875798820

Offset: 1

Seiichi Manyama, Jun 23 2018

nonn

			(1+x)^5*(1+x^2)^15*(1+x^3)^40*(1+x^4)^165* ... = 1 + 5*x + 25*x^2 + 125*x^3 + 625*x^4 + ... .

Seiichi Manyama, Table of n, a(n) for n = 1..1000
Christian G. Bower, PARI programs for transforms, 2007.
N. J. A. Sloane, Maple programs for transforms, 2001-2020.

Inverse Weigh transform of b^n: A306156 (b=2), A306157 (b=3), A306158 (b=4), this sequence (b=5).

Cf. A000351, A038070.

Product_{k>=1} (1+x^k)^a(k) = 1/(1-5x).

a(n) = (1/n) * (5^n + Sum_{d

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A038063 Product_{k>=1}1/(1 - x^k)^a(k) = 1 + 2x.

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A038064 Product_{k>=1} 1/(1 - x^k)^a(k) = 1 + 3x.

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A038065 Product_{k>=1} 1/(1 - x^k)^a(k) = 1 + 4x.

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A038066 Product_{k>=1} 1/(1 - x^k)^a(k) = 1 + 5x.

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A038067 Product_{k>=1} (1 + x^k)^a(k) = 1 + 2x.

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A038068 Product_{k>=1}(1 + x^k)^a(k) = 1 + 3x.

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A038069 Product_{k>=1} ((1 + x^k)^a(k)) = 1 + 4x.

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A306159 Inverse Weigh transform of 5^n.

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