A288851 -id:A288851 - cp's OEIS frontend

A289061 a(n) = 2 * (A288851(n) - 12).

984, 286752, 102360024, 41113157376, 17612599690200, 7859501322760224, 3607454891819189208, 1690291743695465539584, 804566332578533745648600, 387754701670974543569133600, 188763097395728376240220054488

Offset: 1

Seiichi Manyama, Jun 23 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..367

Related to E_{k+2}/E_k: A288995 (k=2), A192731 (k=4), this sequence (k=6).
Cf. A008683, A288840 (E_8*E_6), A288851.
Cf. A289063.

a(n) = (1/n) * Sum_{d|n} A008683(n/d) * A288840(d).

a(n) ~ 2*exp(2*Pi*n) / n. - Vaclav Kotesovec, Jun 03 2018

A289367 a(n) = (2A288851(n) - 3A110163(n))/288.

Original entry on oeis.org

6, 717, 398086, 135369240, 62518201350, 27027759382861, 12577742936206854, 5858597459401083456, 2795780972964509144838, 1345924404035022245534925, 655521004499800309096497414, 321708126100955273726273728024

Offset: 1

Seiichi Manyama, Jul 04 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..368

Cf. A110163, A192731, A288851, A289061, A289365, A289366.

a(n) = (A289061(n) - A192731(n))/288. - Seiichi Manyama, Feb 17 2018

a(n) ~ exp(2*Pi*n) / (144*n). - Vaclav Kotesovec, Jun 03 2018

A289396 a(n) = A288851(n)/12.

Original entry on oeis.org

42, 11949, 4265002, 1713048225, 733858320426, 327479221781677, 150310620492466218, 70428822653977730817, 33523597190772239402026, 16156445902957272648713901, 7865129058155349010009168938, 3860735065245250133098748713633

Offset: 1

Seiichi Manyama, Jul 05 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..367

Cf. A013973 (E_6), A109817 (E_6^(1/12)), A288851.

Cf. A289394, A289395.

a(n) = 1 + (1/(24*n)) * Sum_{d|n} A008683(n/d) * A288840(d).

A299503 a(n) = (1/12) * Sum_{d|n} d * A288851(d).

Original entry on oeis.org

42, 23940, 12795048, 6852216840, 3669291602172, 1964875343509008, 1052174343447263568, 563430581238674063376, 301712374716950167413282, 161564459029576395778765080, 86516419639708839110100858360, 46328820782943003562067180265504

Offset: 1

Seiichi Manyama, Feb 26 2018

nonn

Cf. A013973 (E_6), A109817, A288851, A289540, A300025.

A109817 G.f.: 12th root of Eisenstein series E_6 (cf. A013973).

Original entry on oeis.org

1, -42, -11088, -3774624, -1472710974, -617481728640, -270883381218912, -122585272771463040, -56747118995519331456, -26727350506044696990762, -12760853360973370821796320, -6159994719956314185540737376, -3000691311646502407278581263104, -1472883416501251994527873967792256

Offset: 0

N. J. A. Sloane, Sep 15 2005

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Seiichi Manyama, Table of n, a(n) for n = 0..367
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006.
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

E_6^(k/12): this sequence (k=1), A289325 (k=2), A289326 (k=3), A289327 (k=4), A289328 (k=5), A289293 (k=6), A289345 (k=7), A289346 (k=8), A289347 (k=9), A289348 (k=10), A289349 (k=11).

Cf. A001160, A013973, A288851, A299503, A303055.

nmax = 20; s = 6; CoefficientList[Series[(1 - 2*s/BernoulliB[s] * Sum[DivisorSigma[s - 1, k]*x^k, {k, 1, nmax}])^(1/12), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 02 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(A288851(n)/12). - Seiichi Manyama, Jul 02 2017
a(n) ~ c * exp(2*Pi*n) / n^(13/12), where c = -Gamma(1/4)^(10/3) * Gamma(1/3)^2 / (16 * 6^(1/12) * Pi^3 * Gamma(1/12)) = -0.079329971529325538458906713053582098... - Vaclav Kotesovec, Jul 02 2017, updated Mar 05 2018
Equivalently, c = -Gamma(1/3) * Gamma(1/4)^(7/3) / (2^(23/6) * 3^(11/24) * sqrt(1 + sqrt(3)) * Pi^(5/2)). - Vaclav Kotesovec, Aug 03 2025
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A299503(k)*a(n-k) for n > 0. - Seiichi Manyama, Feb 27 2018
G.f.: Sum_{k>=0} A303055(k) * f(q)^k where f(q) is Sum_{k>=1} sigma_5(k)*q^k. - Seiichi Manyama, Jun 15 2018

A110163 Exponents a(1), a(2), ... such that theta series of E_8 lattice, 1 + 240 q + 2160 q^2 + ... (A004009) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ...

Original entry on oeis.org

-240, 26760, -4096240, 708938760, -130880766192, 25168873498760, -4978357936128240, 1005225129317834760, -206195878414962688240, 42824436296045618358408, -8983966738037593190400240, 1900416270294787067711818760, -404814256845771786255876096240, 86744167089111545378556727322760

Offset: 1

N. J. A. Sloane, Sep 16 2005

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Negative of inverse Euler transform of [240, 2160, ...].

			From _Seiichi Manyama_, Jun 17 2017: (Start)
a(1) = 8 + 1/3 * A008683(1/1) * A288261(1) = 8 + 1/3 * (-744) = -240,
a(2) = 8 + 1/6 * (A008683(2/1) * A288261(1) + A008683(2/2) * A288261(2)) = 8 + 1/6 * (744 + 159768) = 26760. (End)

Seiichi Manyama, Table of n, a(n) for n = 1..424

Cf. A288968 (k=2), this sequence (k=4), A288851 (k=6), A288471 (k=8), A289024 (k=10), A288990/A288989 (k=12), A289029 (k=14).

Cf. A004009, A008683, A013953, A288261, A289636.

terms = 14; Clear[a, sol];
a4009[n_] := If[n == 0, 1, 240 DivisorSigma[3, n]];
sol[0] = {}; sol[kmax_] := sol[kmax] = Join[sol[kmax-1], SolveAlways[ Sum[ a4009[k] q^k, {k, 0, kmax}] == Normal[Product[(1-q^k)^a[k], {k, 1, kmax}] + O[q]^(kmax+1)] /. sol[kmax-1], q][[1]] ];
A110163 = Array[a, terms] /. sol[terms] (* Jean-François Alcover, Jul 03 2017 *)

a(n) = A013953(n^2) for n>=1. - Seiichi Manyama, Jun 17 2017
a(n) = 8 + (1/(3*n)) * Sum_{d|n} A008683(n/d) * A288261(d). - Seiichi Manyama, Jun 17 2017
a(n) = (1/n) * Sum_{d|n} A008683(n/d) * A289636(d). - Seiichi Manyama, Jul 09 2017
a(n) ~ (-1)^n * exp(Pi*sqrt(3)*n) / n. - Vaclav Kotesovec, Mar 08 2018

A289293 Coefficients in expansion of E_6^(1/2).

Original entry on oeis.org

1, -252, -40068, -10158624, -3362961924, -1254502939032, -502480721822688, -211053631376919744, -91717692784641665028, -40892713821496126310364, -18600635229558474625901928, -8597703758971125751979122656

Offset: 0

Seiichi Manyama, Jul 02 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..367

E_k^(1/2): A289291 (k=2), A289292 (k=4), this sequence (k=6), A004009 (k=8), A289294 (k=10), A289295 (k=14).

Cf. A013973 (E_6), A288851.

terms = 12;
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E6[x]^(1/2) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 23 2018 *)

G.f.: Product_{n>=1} (1-q^n)^(A288851(n)/2).

a(n) ~ c * exp(2*Pi*n) / n^(3/2), where c = -3*sqrt(2)*Pi^(3/2) / (16*Gamma(3/4)^8) = -0.2903826839827320330247215149377503818798115... - Vaclav Kotesovec, Jul 02 2017, updated Mar 05 2018

A289326 Coefficients in expansion of E_6^(1/4).

Original entry on oeis.org

1, -126, -27972, -8603784, -3156774138, -1265670056952, -536028623834760, -235629947944839168, -106414175763732002292, -49052892961209924090486, -22977990271885179647877768, -10904016663130642099838196120

Offset: 0

Seiichi Manyama, Jul 02 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..367

E_6^(k/12): A109817 (k=1), A289325 (k=2), this sequence (k=3), A289327 (k=4), A289328 (k=5), A289293 (k=6), A289345 (k=7), A289346 (k=8), A289347 (k=9), A289348 (k=10), A289349 (k=11).

Cf. A013973 (E_6), A288851.

nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^(1/4), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(A288851(n)/4).

a(n) ~ c * exp(2*Pi*n) / n^(5/4), where c = -sqrt(3) * Gamma(1/4)^5 / (32 * 2^(3/4) * Pi^4) = -0.20698746071805886655919194203910626895689130674662074751291... - Vaclav Kotesovec, Jul 08 2017, updated Mar 05 2018

A289325 Coefficients in expansion of E_6^(1/6).

Original entry on oeis.org

1, -84, -20412, -6617856, -2505409788, -1027549673640, -442991672331264, -197605206331169280, -90359564898413083644, -42105781947560460595284, -19913609001700051596476280, -9531377528273693889501019392

Offset: 0

Seiichi Manyama, Jul 02 2017

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			From _Seiichi Manyama_, Jul 08 2017: (Start)
2F1(1/12, 7/12; 1; 1728/(1728 - j))
= 1 - A289557(1)/(j - 1728) + A289557(2)/(j - 1728)^2 - A289557(3)/(j - 1728)^3 + ...
= 1 - 84/(j - 1728) + 62244/(j - 1728)^2 - 64318800/(j - 1728)^3 + ...
= 1 - 84*q - 82656*q^2 -  64795248*q^3 - ...
           + 62244*q^2 + 122496192*q^3 + ...
                       -  64318800*q^3 - ...
                                       + ...
= 1 - 84*q - 20412*q^2 -   6617856*q^3 - ... (End)

Seiichi Manyama, Table of n, a(n) for n = 0..367
R. S. Maier, Nonlinear differential equations satisfied by certain classical modular forms, arXiv:0807.1081 [math.NT], 2008-2010, p. 34 equation (7.30).

E_6^(k/12): A109817 (k=1), this sequence (k=2), A289326 (k=3), A289327 (k=4), A289328 (k=5), A289293 (k=6), A289345 (k=7), A289346 (k=8), A289347 (k=9), A289348 (k=10), A289349 (k=11).

Cf. A013973 (E_6), A288851, A289417, A289557.

nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^(1/6), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(A288851(n)/6).
G.f.: 2F1(1/12, 7/12; 1; 1728/(1728-j)) where j is the elliptic modular invariant (A000521). - Seiichi Manyama, Jul 07 2017
a(n) ~ c * exp(2*Pi*n) / n^(7/6), where c = -Gamma(1/4)^(8/3) * Gamma(1/3)^2 / (2^(9/2) * 3^(1/6) * Pi^(7/2)) = -0.149083170913265334790743918765758886634155... - Vaclav Kotesovec, Jul 08 2017, updated Mar 05 2018

A289327 Coefficients in expansion of E_6^(1/3).

Original entry on oeis.org

1, -168, -33768, -9806496, -3482370024, -1364023149552, -567278132268960, -245678241438057792, -109559333350138970088, -49951945835561166375048, -23173552482577051154061168, -10901813191731667585777068000

Offset: 0

Seiichi Manyama, Jul 02 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..367

E_6^(k/12): A109817 (k=1), A289325 (k=2), A289326 (k=3), this sequence (k=4), A289328 (k=5), A289293 (k=6), A289345 (k=7), A289346 (k=8), A289347 (k=9), A289348 (k=10), A289349 (k=11).

Cf. A013973 (E_6), A288851.

nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^(1/3), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(A288851(n)/3).

a(n) ~ c * exp(2*Pi*n) / n^(4/3), where c = -3^(1/6) * Gamma(1/4)^(16/3) * Gamma(1/3) / (32 * 2^(1/3) * Pi^5) = -0.25096087408563316781920388861983614789... - Vaclav Kotesovec, Jul 08 2017, updated Mar 05 2018

cp's OEIS Frontend

A289061 a(n) = 2 * (A288851(n) - 12).

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A289367 a(n) = (2*A288851(n) - 3*A110163(n))/288.

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A289396 a(n) = A288851(n)/12.

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A299503 a(n) = (1/12) * Sum_{d|n} d * A288851(d).

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A109817 G.f.: 12th root of Eisenstein series E_6 (cf. A013973).

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A110163 Exponents a(1), a(2), ... such that theta series of E_8 lattice, 1 + 240 q + 2160 q^2 + ... (A004009) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ...

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A289293 Coefficients in expansion of E_6^(1/2).

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A289326 Coefficients in expansion of E_6^(1/4).

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A289325 Coefficients in expansion of E_6^(1/6).

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A289327 Coefficients in expansion of E_6^(1/3).

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A289367 a(n) = (2A288851(n) - 3A110163(n))/288.