A110163 -id:A110163 - cp's OEIS frontend

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

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

A289395 a(n) = A110163(n)/8.

Original entry on oeis.org

-30, 3345, -512030, 88617345, -16360095774, 3146109187345, -622294742016030, 125653141164729345, -25774484801870336030, 5353054537005702294801, -1122995842254699148800030, 237552033786848383463977345, -50601782105721473281984512030

Offset: 1

Seiichi Manyama, Jul 05 2017

sign

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

Cf. A004009 (E_4), A108091 (E_4^(1/8)), A110163.

Cf. A289394, A289396.

a(n) = 1 + (1/(24*n)) * Sum_{d|n} A008683(n/d) * A288261(d).

A300147 a(n) = (1/8) * Sum_{d|n} d * A110163(d).

Original entry on oeis.org

-30, 6660, -1536120, 354476040, -81800478900, 18876653594640, -4356063194112240, 1005225129672310800, -231970363216834560390, 53530545369975222475800, -12352954264801690636800360, 2850624405442199478575792160

Offset: 1

Seiichi Manyama, Feb 26 2018

sign

Vaclav Kotesovec, Table of n, a(n) for n = 1..420

Cf. A004009 (E_4), A108091, A110163, A289247, A300025.

a(n) ~ (-1)^n * exp(Pi*sqrt(3)*n) / 8. - Vaclav Kotesovec, Jun 07 2018

A289633 a(n) = 6 * Sum_{d|n} d * A110163(d).

Original entry on oeis.org

-1440, 319680, -73733760, 17014849920, -3926422987200, 906079372542720, -209091033317387520, 48250806224270918400, -11134577434408058898720, 2569466177758810678838400, -592941804710481150566417280, 136829971461225574971638023680

Offset: 1

Seiichi Manyama, Jul 08 2017

sign

			G.f.: -1440*q + 319680*q^2 - 73733760*q^3 + 17014849920*q^4 - 3926422987200*q^5 + ...
a(1) = 6 * (1 * A110163(1)) = -1440,
a(2) = 6 * (1 * A110163(1) + 2 * A110163(2)) = 319680,
a(3) = 6 * (1 * A110163(1) + 3 * A110163(3)) = -73733760.

Seiichi Manyama, Table of n, a(n) for n = 1..422
Ken Ono, The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-series, CBMS Regional Conference Series in Mathematics, vol. 102, American Mathematical Society, Providence, RI, 2004.

Cf. A000594, A110163, A126839 (A000594(n) mod 11), A289636.

a(n) == A000594(n) mod 11.

a(n) ~ 6 * (-1)^n * exp(Pi*sqrt(3)*n). - Vaclav Kotesovec, Jul 09 2017

A108091 Coefficients of series whose 8th power is the theta series of E_8 (see A004009).

Original entry on oeis.org

1, 30, -2880, 416640, -69178110, 12378401280, -2321610157440, 449733567736320, -89200812128140800, 18013245273252679710, -3689479088922151082880, 764375901202388789804160, -159862757100127037505991680, 33699694000689939789618455040, -7152050326608893289997995966720, 1526705794390267864554876727856640

Offset: 0

N. J. A. Sloane and Michael Somos, Jun 06 2005

sign

			More precisely, the theta series of E_8 begins 1 + 240*q^2 + 2160*q^4 + 6720*q^6 + 17520*q^8 + ... and the 8th root of this is 1 + 30*q^2 - 2880*q^4 + 416640*q^6 - 69178110*q^8 + ...

N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.

Seiichi Manyama, Table of n, a(n) for n = 0..424
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.
N. J. A. Sloane, Seven Staggering Sequences.

E_4^(k/8): this sequence (k=1), A289307 (k=2), A289308 (k=3), A289292 (k=4), A289309 (k=5).

Cf. A001158, A004009 (E_4), A106205, A110163, A300147, A303007.

nmax = 20; s = 8; CoefficientList[Series[(1 - 2*s/BernoulliB[s] * Sum[DivisorSigma[s - 1, k]*x^k, {k, 1, nmax}])^(1/16), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 02 2017 *)

R. = PowerSeriesRing(ZZ,20)
a = R(eisenstein_series_qexp(4,20, normalization='integral'))
list(a.sqrt().sqrt().sqrt()) # Andy Huchala, Jul 10 2021

G.f.: Product_{n>=1} (1-q^n)^(A110163(n)/8). - Seiichi Manyama, Jul 02 2017
a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(9/8), where c = 3^(1/4) * Gamma(1/3)^(9/4) / (2^(33/8) * Pi^(3/2) * Gamma(7/8)) = 0.1141392450598624077174159151600898926678394937157356242319309115... - Vaclav Kotesovec, Jul 02 2017, updated Mar 05 2018
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A300147(k)*a(n-k) for n > 0. - Seiichi Manyama, Feb 27 2018
G.f.: Sum_{k>=0} A303007(k) * (-f(q))^k where f(q) is Sum_{k>=1} sigma_3(k)*q^k. - Seiichi Manyama, Jun 15 2018

A288851 Exponents a(1), a(2), ... such that E_6, 1 - 504q - 16632q^2 - ... (A013973) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

Original entry on oeis.org

504, 143388, 51180024, 20556578700, 8806299845112, 3929750661380124, 1803727445909594616, 845145871847732769804, 402283166289266872824312, 193877350835487271784566812, 94381548697864188120110027256, 46328820782943001597184984563596

Offset: 1

Seiichi Manyama, Jun 18 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..367
R. E. Borcherds, Automorphic forms on O_{s+2,2}(R)^{+} and generalized Kac-Moody algebras, pp. 744-752 of Proc. Intern. Congr. Math., Vol. 2, 1994.

Cf. A288968 (k=2), A110163 (k=4), this sequence (k=6), A288471 (k=8), A289024 (k=10), A288990/A288989 (k=12), A289029 (k=14).

Cf. A008683, A013973 (E_6), A110163, A288840 (E_8/E_6), A289637.

a(n) = A013975(n^2) for n>=1.
a(n) = 12 + (1/(2*n)) * Sum_{d|n} A008683(n/d) * A288840(d).
a(n) = (1/n) * Sum_{d|n} A008683(n/d) * A289637(d). - Seiichi Manyama, Jul 09 2017
a(n) ~ exp(2*Pi*n) / n. - Vaclav Kotesovec, Mar 08 2018

A192731 Euler transform is 1 / (q j(q)) where j is j-function (A000521).

Original entry on oeis.org

-744, 80256, -12288744, 2126816256, -392642298600, 75506620496256, -14935073808384744, 3015675387953504256, -618587635244888064744, 128473308888136855075200, -26951900214112779571200744

Offset: 1

Michael Somos, Jul 08 2011

sign

			From _Seiichi Manyama_, Jun 18 2017: (Start)
a(1) = (1/1) * A008683(1/1) * A288261(1) = (1/1) * (-744) = -744,
a(2) = (1/2) * (A008683(2/1) * A288261(1) + A008683(2/2) * A288261(2)) = (1/2) * (744 + 159768) = 80256. (End)

Seiichi Manyama, Table of n, a(n) for n = 1..424
B. Brent, p-adic continuity for exponents in product decomposition of the j-invariant, Answer 3 by W. Zudilin
N. J. A. Sloane, Transforms

Cf. A008683, A063995, A110163, A192732, A288261, A302407, A305757.

{a(n) = local(A, S); if( n<1, 0, A = 1 + x * O(x^n); S = x * ellj( x * A ); for( k = 1, n-1, S *= (A - x^k) ^ polcoeff( S, k)); - polcoeff( S, n))}

1 / (q j(q)) = Product_{k>0} (1 - x^k)^-a(k).
a(n) = 3*(A110163(n) - 8) = (1/n) * Sum_{d|n} A008683(n/d) * A288261(d). - Seiichi Manyama, Jun 18 2017
a(n) ~ (-1)^n * 3*exp(Pi*sqrt(3)*n) / n. - Vaclav Kotesovec, Mar 24 2018

A288261 Coefficients in expansion of E_6/E_4.

Original entry on oeis.org

1, -744, 159768, -36866976, 8507424792, -1963211493744, 453039686271072, -104545516658693952, 24125403112135458840, -5567288717204029449672, 1284733088879405339418768, -296470902355240575283208928, 68414985730612787485819011168

Offset: 0

Seiichi Manyama, Jun 17 2017

sign

Also coefficients in expansion of E_10/E_8. - Seiichi Manyama, Jun 20 2017

			G.f.: 1 - 744*q + 159768*q^2 - 36866976*q^3 + 8507424792*q^4 - 1963211493744*q^5 + 453039686271072*q^6 + ...
From _Seiichi Manyama_, Jun 27 2017: (Start)
a(0) = j_0((-1+sqrt(3)*i)/2) = 1,_
a(1) = j_1((-1+sqrt(3)*i)/2) = -744 + 0^1 = -744,
a(2) = j_2((-1+sqrt(3)*i)/2) = 159768 - 1488*0^1 + 0^2 = 159768. (End)

Seiichi Manyama, Table of n, a(n) for n = 0..422

Cf. A004009 (E_4), A110163, A013973 (E_6).
E_{k+2}/E_k: A288877 (k=2), this sequence (k=4, 8), A288840 (k=6).
Cf. A000521 (j), A035230 (-q*j'), A066395 (1/j), A289141.

nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5, k]*x^k, {k, 1, nmax}])/(1 + 240*Sum[DivisorSigma[3, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 28 2017 *)
terms = 13; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[Ei[6]/Ei[4] + O[x]^terms, x] (* Jean-François Alcover, Mar 01 2018 *)
a[ n_] := With[{j = Series[1728 KleinInvariantJ[ Log[ Series[q, {q, 0, n + 1}]]/(2 Pi I)], {q, 0, n}]}, SeriesCoefficient[ -q D[j, q] / j, {q, 0, n}]]; (* Michael Somos, Aug 15 2018 *)

From Seiichi Manyama, Jun 27 2017: (Start)
Let j_0 = 1 and j_1 = j - 744. Define j_m by j_m = j1 | T_0(m), where T_0(m) = mT_{m, 0} is the normalized m-th weight zero Hecke operator. a(n) = j_n((-1+sqrt(3)*i)/2).
G.f.: Sum_{n >= 0} j_n((-1+sqrt(3)*i)/2)*q^n. (End)
a(n) ~ (-1)^n * 3 * exp(Pi*sqrt(3)*n). - Vaclav Kotesovec, Jun 28 2017
G.f.: -q*j'/j where j is the elliptic modular invariant (A000521). - Seiichi Manyama, Jul 12 2017

A289292 Coefficients in expansion of E_4^(1/2).

Original entry on oeis.org

1, 120, -6120, 737760, -107249640, 17385063120, -3014720249760, 547287510713280, -102701836021530600, 19762301660609250840, -3878226140959368843120, 773209219953012480001440, -156173318001506652330786720, 31888935085481430265623676560

Offset: 0

Seiichi Manyama, Jul 02 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..425
R. S. Maier, Nonlinear differential equations satisfied by certain classical modular forms, arXiv:0807.1081 [math.NT], 2008-2010, p. 34 equation (7.29b).

E_k^(1/2): A289291 (k=2), this sequence (k=4), A289293 (k=6), A004009 (k=8), A289294 (k=10), A289295 (k=14).
E_4^(k/8): A108091 (k=1), A289307 (k=2), A289308 (k=3), this sequence (k=4), A289309 (k=5), A289318 (k=6), A289319 (k=7).
Cf. A001421, A004009 (E_4), A110163.

terms = 14;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E4[x]^(1/2) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 23 2018 *)

G.f.: Product_{n>=1} (1-q^n)^(A110163(n)/2).
a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(3/2), where c = 3*Gamma(1/3)^9 / (32*sqrt(2)*Pi^(13/2)) = 0.27646925986847687648926173728588572192308632719... - Vaclav Kotesovec, Jul 02 2017, updated Mar 05 2018
G.f.: 3F2(1/6, 1/2, 5/6; 1, 1; 1728/j) where j is the elliptic modular invariant (A000521). - Seiichi Manyama, Jul 07 2017

A289307 Coefficients in expansion of E_4^(1/4) in powers of q.

Original entry on oeis.org

1, 60, -4860, 660480, -105063420, 18206269560, -3328461434880, 631226199152640, -122944850563477500, 24436796345920143420, -4935178772322020730360, 1009598430837232126725120, -208736157503462405753487360, 43541664791244563211024015480

Offset: 0

Seiichi Manyama, Jul 02 2017

sign

			From _Seiichi Manyama_, Jul 07 2017: (Start)
2F1(1/12, 5/12; 1; 1728/j)
= 1 + (1*5)/(1*1) * 12/j + (1*5*13*17)/(1*1*2*2) * (12/j)^2 + (1*5*13*17*25*29)/(1*1*2*2*3*3) * (12/j)^3 + ...
= 1 + 60/j + 39780/j^2 + 38454000/j^3 + ...
= 1 + 60*q - 44640*q^2 + 21399120*q^3 - ...
           + 39780*q^2 - 59192640*q^3 + ...
                       + 38454000*q^3 - ...
                                      + ...
= 1 + 60*q -  4860*q^2 +   660480*q^3 - ... (End)

Seiichi Manyama, Table of n, a(n) for n = 0..424
M. Kontsevich and D. Zagier, Periods, Institut des Hautes Etudes Scientifiques 2001 IHES/M/01/22. Published in B. Engquist and W. Schmid, editors, Mathematics Unlimited - 2001 and Beyond, 2 vols., Springer-Verlag, 2001, pp. 771-808, section 2.3. Example 3.
R. S. Maier, Nonlinear differential equations satisfied by certain classical modular forms, arXiv:0807.1081 [math.NT], 2008-2010, p. 34 equation (7.29a).

E_4^(k/8): A108091 (k=1), this sequence (k=2), A289308 (k=3), A289292 (k=4), A289309 (k=5), A289318 (k=6), A289319 (k=7).

Cf. A000521 (j), A004009 (E_4), A066395 (1/j), A092870, A110163, A289210.

a[ n_] := SeriesCoefficient[ ComposeSeries[ Series[ Hypergeometric2F1[ 1/12, 5/12, 1, q], {q, 0, n}], q^2 / Series[q^2 KleinInvariantJ[ Log[q]/(2 Pi I)], {q, 0, n}]], {q, 0, n}]; (* Michael Somos, Jun 21 2018 *)

G.f.: Product_{n>=1} (1-q^n)^(A110163(n)/4).
G.f.: 2F1(1/12, 5/12; 1; 1728/j) where j is the elliptic modular invariant (A000521). - Seiichi Manyama, Jul 06 2017 [See also the Kontsevich and Zagier link, where t = 1728/j = 1 - Sum_{k>=0} A289210(k)*q^k, with q = q(z) = exp(2*Pi*I*z), Im(z) > 0. - Wolfdieter Lang, May 27 2018]
a(n) ~ (-1)^(n+1) * c * exp(Pi*sqrt(3)*n) / n^(5/4), where c = sqrt(3) * Gamma(1/3)^(9/2) * Gamma(1/4) / (16 * 2^(3/4) * Pi^4) = 0.201967785736579402060958871696381229013432952780653381728912717635... - Vaclav Kotesovec, Jul 07 2017, updated Mar 04 2018

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Formula

A289395 a(n) = A110163(n)/8.

Views

Author

Keywords

Links

Crossrefs

Formula

A300147 a(n) = (1/8) * Sum_{d|n} d * A110163(d).

Views

Author

Keywords

Links

Crossrefs

Formula

A289633 a(n) = 6 * Sum_{d|n} d * A110163(d).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A108091 Coefficients of series whose 8th power is the theta series of E_8 (see A004009).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Sage

Formula

A288851 Exponents a(1), a(2), ... such that E_6, 1 - 504*q - 16632*q^2 - ... (A013973) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

Views

Author

Keywords

Links

Crossrefs

Formula

A192731 Euler transform is 1 / (q j(q)) where j is j-function (A000521).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A288261 Coefficients in expansion of E_6/E_4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A289292 Coefficients in expansion of E_4^(1/2).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A289307 Coefficients in expansion of E_4^(1/4) in powers of q.

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

A288851 Exponents a(1), a(2), ... such that E_6, 1 - 504q - 16632q^2 - ... (A013973) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .