A289417 -id:A289417 - cp's OEIS frontend

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

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

A106203 Coefficients of ((j(q)-1728)q)^(1/24) where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, -41, -11128, -3785793, -1476507895, -618962022329, -271503819749095, -122857395553223337, -56870247894888518054, -26784343611333662213130, -12787694574831980406719382, -6172809198874485994313412898

Offset: 0

Michael Somos, Apr 25 2005

sign

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

(q*(j(q)-1728))^(k/24): A289563 (k=-96), A289562 (k=-72), A289561 (k=-48), A289417 (k=-24), A289416 (k=-1), this sequence (k=1), A289330 (k=2), A289331 (k=3), A289332 (k=4), A289333 (k=5), A289334 (k=6), A007242 (k=12), A289063 (k=24).

Cf. A000521, A289061.

CoefficientList[Series[((256/QPochhammer[-1, x]^8 + x*QPochhammer[-1, x]^16/256)^3 - 1728*x)^(1/24), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 07 2018 *)

{a(n)=if(n<0,0, polcoeff( ((ellj(x+x^2*O(x^n))-1728)*x)^(1/24),n))}

G.f.: Product_{k>=1} (1-q^k)^(A289061(k)/24). - Seiichi Manyama, Jul 02 2017

a(n) ~ c * exp(2*Pi*n) / n^(13/12), where c = -2^(1/12) * Pi^(25/12) * exp(-Pi/12) / (3^(13/12) * Gamma(2/3)^2 * Gamma(3/4)^(7/3) * Gamma(1/12)) = -0.0794786705643291777786030631826408355507134016936764993676699378963... - Vaclav Kotesovec, Mar 07 2018

A288840 Coefficients in expansion of E_8/E_6.

Original entry on oeis.org

1, 984, 574488, 307081056, 164453203992, 88062998451984, 47157008244215904, 25252184242734325440, 13522333949728177520664, 7241096993206804017918456, 3877547016709833498690361488, 2076394071353012138642420600352

Offset: 0

Seiichi Manyama, Jun 17 2017

nonn

			G.f.: 1 + 984*q + 574488*q^2 + 307081056*q^3 + 164453203992*q^4 + 88062998451984*q^5 + 47157008244215904*q^6 + ...
From _Seiichi Manyama_, Jun 27 2017: (Start)
a(0) = j_0(i) = 1,_
a(1) = j_1(i) = -744 + 1728^1 = 984,
a(2) = j_2(i) = 159768 - 1488*1728^1 + 1728^2 = 574488. (End)

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.

Seiichi Manyama, Table of n, a(n) for n = 0..366

Cf. A013973 (E_6), A008410 (E_8).
Cf. A288261 (E_6/E_4).
Cf. A000521 (j), A035230 (-q*j'), A289141, A289417.

nmax = 20; CoefficientList[Series[(1 + 480*Sum[DivisorSigma[7, k]*x^k, {k, 1, nmax}])/(1 - 504*Sum[DivisorSigma[5, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 28 2017 *)
terms = 12; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[Ei[8]/Ei[6] + O[x]^terms, x] (* Jean-François Alcover, Mar 01 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(i).
G.f.: Sum_{n >= 0} j_n(i)*q^n. (End)
a(n) ~ 2 * exp(2*Pi*n). - Vaclav Kotesovec, Jun 28 2017
G.f.: -q*j'/(j-1728) where j is the elliptic modular invariant (A000521). - Seiichi Manyama, Jul 12 2017

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

sign

			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

A289416 Coefficients of (q*(j(q)-1728))^(-1/24) where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, 41, 12809, 4767210, 1969719570, 861799083811, 391094324350380, 182038077972154741, 86322373755372340110, 41521193849940130872000, 20197774625594843441436930, 9915082544034345319047507780

Offset: 0

Seiichi Manyama, Jul 06 2017

nonn

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

(q*(j(q)-1728))^(k/24): A289563 (k=-96), A289562 (k=-72), A289561 (k=-48), A289417 (k=-24), this sequence (k=-1), A106203 (k=1), A289330 (k=2), A289331 (k=3), A289332 (k=4), A289333 (k=5), A289334 (k=6), A007242 (k=12), A289063 (k=24).

CoefficientList[Series[((256/QPochhammer[-1, x]^8 + x*QPochhammer[-1, x]^16/256)^3 - 1728*x)^(-1/24), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 07 2018 *)

G.f.: Product_{n>=1} (1-q^n)^(-A289061(n)/24) = Product_{n>=1} (1-q^n)^(1-A289396(n)).

a(n) ~ c * exp(2*Pi*n) / n^(11/12), where c = Gamma(3/4)^(1/3) * exp(Pi/12) / (2^(1/12) * 3^(1/6) * Pi^(1/12) * Gamma(1/12)) = 0.086380262154841817375196725... - Vaclav Kotesovec, Mar 07 2018

A289561 Coefficients of 1/(q*(j(q)-1728))^2 where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, 1968, 2511000, 2605664960, 2387651205420, 2011663789279200, 1594903822090229312, 1207416525204065938560, 881461062200198781904590, 624887481909094711741279120, 432393768184906363401468637728, 293171504960988659691658645670592

Offset: 0

Seiichi Manyama, Jul 08 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..363

(q*(j(q)-1728))^(k/24): A289563 (k=-96), A289562 (k=-72), this sequence (k=-48), A289417 (k=-24), A289416 (k=-1), A106203 (k=1), A289330 (k=2), A289331 (k=3), A289332 (k=4), A289333 (k=5), A289334 (k=6), A007242 (k=12), A289063 (k=24).

Cf. A289061.

CoefficientList[Series[((256/QPochhammer[-1, x]^8 + x*QPochhammer[-1, x]^16/256)^3 - 1728*x)^(-2), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 07 2018 *)

G.f.: Product_{n>=1} (1-q^n)^(-2*A289061(n)).

a(n) ~ c * exp(2*Pi*n) * n^3, where c = Gamma(3/4)^16 * exp(4*Pi) / (629856 * Pi^4) = 0.120838515551739021017044909469013807578104459775498957232984908667972... - Vaclav Kotesovec, Mar 07 2018

A289562 Coefficients of 1/(q*(j(q)-1728))^3 where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, 2952, 5218884, 7138351488, 8319960432666, 8678332561127616, 8338315178481134040, 7518590274496806176256, 6444205834302869333758299, 5298802621872639665867604832, 4208666443076672300677008045636, 3246069554930472099322915758511872

Offset: 0

Seiichi Manyama, Jul 08 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..362

(q*(j(q)-1728))^(k/24): A289563 (k=-96), this sequence (k=-72), A289561 (k=-48), A289417 (k=-24), A289416 (k=-1), A106203 (k=1), A289330 (k=2), A289331 (k=3), A289332 (k=4), A289333 (k=5), A289334 (k=6), A007242 (k=12), A289063 (k=24).

Cf. A289061.

CoefficientList[Series[((256/QPochhammer[-1, x]^8 + x*QPochhammer[-1, x]^16/256)^3 - 1728*x)^(-3), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 07 2018 *)

G.f.: Product_{n>=1} (1-q^n)^(-3*A289061(n)).

a(n) ~ c * exp(2*Pi*n) * n^5, where c = Gamma(3/4)^24 * exp(6*Pi) / (4081466880 * Pi^6) = 0.0051446247390864841578336638645072392120317488530740050289688... - Vaclav Kotesovec, Mar 07 2018

A289563 Coefficients of 1/(q*(j(q)-1728))^4 where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, 3936, 8895024, 15094625920, 21336320693400, 26506772152211520, 29887990556174431424, 31237788209244729015552, 30709242534935581933885740, 28700724444538653431660487520, 25706227251014342788669659769056, 22202613798662970539127791744222592

Offset: 0

Seiichi Manyama, Jul 08 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..361

(q*(j(q)-1728))^(k/24): this sequence (k=-96), A289562 (k=-72), A289561 (k=-48), A289417 (k=-24), A289416 (k=-1), A106203 (k=1), A289330 (k=2), A289331 (k=3), A289332 (k=4), A289333 (k=5), A289334 (k=6), A007242 (k=12), A289063 (k=24).

Cf. A289061.

CoefficientList[Series[((256/QPochhammer[-1, x]^8 + x*QPochhammer[-1, x]^16/256)^3 - 1728*x)^(-4), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 07 2018 *)

G.f.: Product_{n>=1} (1-q^n)^(-4*A289061(n)).

a(n) ~ c * exp(2*Pi*n) * n^7, where c = Gamma(3/4)^32 * exp(8*Pi) / (55540601303040 * Pi^8) = 0.0001042996202910562374208781457852661312263780276025385904... - Vaclav Kotesovec, Mar 07 2018

A305699 Coefficients of 1/(q*(j(q)-744)) where j(q) is the elliptic modular invariant.

Original entry on oeis.org

1, 0, -196884, -21493760, 37899009486, 8443309031424, -6829893232051144, -2454385780209696768, 1130962845597176786661, 621972524796731658731520, -164194903359722124902384028, -144508453392903668301846454272

Offset: 0

Seiichi Manyama, Jun 08 2018

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..370
Eric Weisstein's World of Mathematics, j-Function

(q*(j(q)-744))^(k/4): A305698 (k=-2), A305696 (k=-1), A304020 (k=1), A305697 (k=2).

Cf. A000521 (j), A014708 (j-744), A066395, A289417.

CoefficientList[Series[1/((2^16 + x*QPochhammer[-1, x]^24)^3/(2*QPochhammer[-1, x])^24 - 744*x), {x, 0, 15}], x] (* Vaclav Kotesovec, Jun 09 2018 *)

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

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A106203 Coefficients of ((j(q)-1728)q)^(1/24) where j(q) is the elliptic modular invariant.

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A288840 Coefficients in expansion of E_8/E_6.

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

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A289416 Coefficients of (q*(j(q)-1728))^(-1/24) where j(q) is the elliptic modular invariant.

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A289561 Coefficients of 1/(q*(j(q)-1728))^2 where j(q) is the elliptic modular invariant.

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A289562 Coefficients of 1/(q*(j(q)-1728))^3 where j(q) is the elliptic modular invariant.

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A289563 Coefficients of 1/(q*(j(q)-1728))^4 where j(q) is the elliptic modular invariant.

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A305699 Coefficients of 1/(q*(j(q)-744)) where j(q) is the elliptic modular invariant.

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