A288840 -id:A288840 - cp's OEIS frontend

A294975 a(n) = (1/(24n)) Sum_{d|n} A008683(n/d) * (A288840(d) - A288877(d)).

30, 11775, 4261790, 1712983575, 733856931102, 327479190724415, 150310619778297630, 70428822637214055855, 33523597190372498303390, 16156445902947621421555071, 7865129058155113639991368350, 3860735065245244345161225213335

Offset: 1

Seiichi Manyama, Feb 12 2018

nonn

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

Cf. A008683, A288840, A288877, A294976.

terms = 12;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E8[x_] = 1 + 480*Sum[k^7*x^k/(1 - x^k), {k, 1, terms}];
a[n_] := (1/(24 n))*Sum[MoebiusMu[n/d]*SeriesCoefficient[E8[x]/E6[x] - E4[x]/E2[x], {x, 0, d}], {d, Divisors[n]}];
Array[a, terms] (* Jean-François Alcover, Feb 26 2018 *)

a(n) ~ exp(2*Pi*n) / (12*n). - Vaclav Kotesovec, Jun 03 2018

A307760 a(n) = (A288840(5*n) - A288840(n))/3000.

Original entry on oeis.org

0, 29354332817, 1292515672236611166229929, 56911411397043798759062515920346474052, 2505895144465774474827581824222076409739506505294889, 110338336739674093788639219775047689525384763850603917156033805525

Offset: 0

Seiichi Manyama, Apr 26 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..73
MathOverflow, Properties of coefficients in expansion of E6/E4 and E8/E6

Cf. A008410 (E_8), A013973 (E_6), A288840, A307759.

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

A289209 Coefficients in expansion of E_4^3/E_6^2.

Original entry on oeis.org

1, 1728, 1700352, 1332930816, 939690602496, 624182333927040, 399031077924476928, 248370528839869094400, 151578005556161702559744, 91116938989182168182098368, 54119528875319902426524072960, 31833210323194251819350736777984

Offset: 0

Seiichi Manyama, Jun 28 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..365

(E_4^3/E_6^2)^(k/288): A289365 (k=1), A299694 (k=2), A299696 (k=3), A299697 (k=4), A299698 (k=6), A299943 (k=8), A299949 (k=9), A289369 (k=12), A299950 (k=16), A299951 (k=18), A299953 (k=24), A299993 (k=32), A299994 (k=36), A300052 (k=48), A300053 (k=72), A300054 (k=96), A300055 (k=144), this sequence (k=288).
Cf. A004009 (E_4), A013973 (E_6), A289063, A289210, A289417, A300025.
E_{k+2}/E_k: A288261 (k=4, 8), A288840 (k=6).

nmax = 20; CoefficientList[Series[(1 + 240*Sum[DivisorSigma[3,k]*x^k, {k, 1, nmax}])^3 / (1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^2, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

G.f.: 1 + 1728 * q * Product_{k>=1} (1-q^k)^24 / E_6^2.
G.f.: (E_4*E_8)/(E_6*E_6) = (E_8*E_8)/(E_6*E_10). - Seiichi Manyama, Jun 29 2017
a(n) = 1728 * A289417(n - 1) for n > 0. - Seiichi Manyama, Jul 08 2017
a(n) ~ c * exp(2*Pi*n) * n, where c = 256 * Pi^6 / (3 * Gamma(1/4)^8) = 2.747700206704861755142526128354171788550012833617513654955480535522... - Vaclav Kotesovec, Jul 08 2017, updated Mar 04 2018
a(0) = 1, a(n) = (288/n)*Sum_{k=1..n} A300025(k)*a(n-k) for n > 0. - Seiichi Manyama, Feb 26 2018

A289210 Coefficients in expansion of E_6^2/E_4^3.

Original entry on oeis.org

1, -1728, 1285632, -616294656, 242544070656, -85253786824320, 27846073156184064, -8638345400999827968, 2579332695698905989120, -747814048389765750131136, 211795259563761765262894080, -58852853362216364363212075776

Offset: 0

Seiichi Manyama, Jun 28 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..420

(E_6^2/E_4^3)^(k/288): A289366 (k=1), A296609 (k=2), A296614 (k=3), A296652 (k=4), A297021 (k=6), A299422 (k=8), A299862 (k=9), A289368 (k=12), A299856 (k=16), A299857 (k=18), A299858 (k=24), A299863 (k=32), A299859 (k=36), A299860 (k=48), A299861 (k=72), A299414 (k=96), A299413 (k=144), this sequence (k=288).
Cf. A000521 (j), A004009 (E_4), A013973 (E_6), A066395, A289209, A300025.
E_{k+2}/E_k: A288261 (k=4, 8), A288840 (k=6).

nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^2 / (1 + 240*Sum[DivisorSigma[3,k]*x^k, {k, 1, nmax}])^3, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

a(n) = -1728 * A066395(n) for n > 0.
G.f.: 1 - 1728 * q * Product_{k>=1} (1-q^k)^24 / E_4^3 = 1 - 1728/j.
G.f.: (E_6*E_6)/(E_4*E_8) = (E_6*E_10)/(E_8*E_8). - Seiichi Manyama, Jun 29 2017
a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) * n^2, where c = 256 * Pi^12 / Gamma(1/3)^18 = 4.684993039417145659090436569582265840407909701042523126716193567422... - Vaclav Kotesovec, Jul 08 2017, updated Mar 04 2018
a(0) = 1, a(n) = -(288/n)*Sum_{k=1..n} A300025(k)*a(n-k) for n > 0. - Seiichi Manyama, Feb 26 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

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

Original entry on oeis.org

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

A288877 Coefficients in expansion of E_4/E_2.

Original entry on oeis.org

1, 264, 8568, 231456, 6214872, 166719024, 4472485344, 119980322880, 3218631807384, 86344077536616, 2316294684846288, 62137684699355232, 1666926011246777184, 44717506621139113584, 1199606572169515887552, 32181041313068138778816

Offset: 0

Seiichi Manyama, Jun 18 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..699

E_{k+2}/E_k: this sequence (k=2), A288261 (k=4), A288840 (k=6).
Cf. A004009 (E_4), A006352 (E_2), A288816 (1/E_2).
Cf. A211342.

nmax = 20; CoefficientList[Series[(1 + 240*Sum[DivisorSigma[3, k]*x^k, {k, 1, nmax}])/(1 - 24*Sum[DivisorSigma[1, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 28 2017 *)
terms = 16; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[Ei[4]/Ei[2] + O[x]^terms, x] (* Jean-François Alcover, Mar 01 2018 *)

a(n) ~ 12 / r^n, where r = A211342 = 0.037276810296451658150980785651644618... is the root of the equation Sum_{k>=1} A000203(k) * r^k = 1/24. - Vaclav Kotesovec, Jun 28 2017

A289029 Exponents a(1), a(2), ... such that E_14, 1 - 24q - 196632q^2 + ... (A058550) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

Original entry on oeis.org

24, 196908, 42987544, 21974456220, 8544538312728, 3980088408377644, 1793770730037338136, 847156322106368439324, 401870774532436947447832, 193962999708079363021283628, 94363580764388112933729226776, 46332621615483591171320408201116

Offset: 1

Seiichi Manyama, Jun 22 2017

nonn

This sequence is related to the identity: E_4^2*E_6 = E_4*E_10 = E_6*E_8 = E_14.

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

Cf. A288968 (k=2), A110163 (k=4), A288851 (k=6), A288471 (k=8), A289024 (k=10), A288990/A288989 (k=12), this sequence (k=14).

Cf. A008683, A288261 (E_6/E_4), A288840 (E_8/E_6), A289640.

a(n) = 2 * A110163(n) + A288851(n) = A110163(n) + A289024(n) = A288851(n) + A288471(n) = 28 + (1/n) * (Sum_{d|n} A008683(n/d) * (2/3 * A288261(d) + 1/2 * A288840(d))).
a(n) = (1/n) * Sum_{d|n} A008683(n/d) * A289640(d). - Seiichi Manyama, Jul 09 2017
a(n) ~ exp(2*Pi*n) / n. - Vaclav Kotesovec, Mar 08 2018

A289024 Exponents a(1), a(2), ... such that E_10, 1 - 264q - 135432q^2 + ... (A013974) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

Original entry on oeis.org

264, 170148, 47083784, 21265517460, 8675419078920, 3954919534878884, 1798749087973466376, 846151096977050604564, 402076970410851910136072, 193920175271783317402925220, 94372564731126150526919627016, 46330721199213296384252696382356

Offset: 1

Seiichi Manyama, Jun 22 2017

nonn

This sequence is related to the identity: E_4*E_6 = E_10.

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

Cf. A288968 (k=2), A110163 (k=4), A288851 (k=6), A288471 (k=8), this sequence (k=10), A288990/A288989 (k=12), A289029 (k=14).

Cf. A008683, A288261 (E_6/E_4), A288840 (E_8/E_6), A289639.

a(n) = A110163(n) + A288851(n) = 20 + (1/n) * (Sum_{d|n} A008683(n/d) * (1/3 * A288261(d) + 1/2 * A288840(d))).
a(n) = (1/n) * Sum_{d|n} A008683(n/d) * A289639(d). - Seiichi Manyama, Jul 09 2017
a(n) ~ exp(2*Pi*n) / n. - Vaclav Kotesovec, Mar 08 2018

cp's OEIS Frontend

A294975 a(n) = (1/(24*n)) * Sum_{d|n} A008683(n/d) * (A288840(d) - A288877(d)).

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A307760 a(n) = (A288840(5*n) - A288840(n))/3000.

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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) ... .

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A289209 Coefficients in expansion of E_4^3/E_6^2.

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A289210 Coefficients in expansion of E_6^2/E_4^3.

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A288261 Coefficients in expansion of E_6/E_4.

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

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A288877 Coefficients in expansion of E_4/E_2.

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A289029 Exponents a(1), a(2), ... such that E_14, 1 - 24*q - 196632*q^2 + ... (A058550) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

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A289024 Exponents a(1), a(2), ... such that E_10, 1 - 264*q - 135432*q^2 + ... (A013974) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

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A294975 a(n) = (1/(24n)) Sum_{d|n} A008683(n/d) * (A288840(d) - A288877(d)).

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) ... .

A289029 Exponents a(1), a(2), ... such that E_14, 1 - 24q - 196632q^2 + ... (A058550) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

A289024 Exponents a(1), a(2), ... such that E_10, 1 - 264q - 135432q^2 + ... (A013974) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .