cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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

Views

Author

Seiichi Manyama, Jun 17 2017

Keywords

Comments

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

Examples

			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)

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)

Formula

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

A289063 Coefficients in expansion of E_6^2/Product_{k>=1} (1-q^k)^24.

Original entry on oeis.org

1, -984, 196884, 21493760, 864299970, 20245856256, 333202640600, 4252023300096, 44656994071935, 401490886656000, 3176440229784420, 22567393309593600, 146211911499519294, 874313719685775360, 4872010111798142520, 25497827389410525184, 126142916465781843075
Offset: 0

Views

Author

Seiichi Manyama, Jun 23 2017

Keywords

Comments

Convolution square of A007242. - Michael Somos, Mar 31 2019

Examples

			G.f. = (1-q)^984 * (1-q^2)^286752 * (1-q^3)^102360024 * ...
G.f. = 1 - 984*q + 196884*q^2 + 21493760*q^3 + 864299970*q^4 + 20245856256*q^5 + ... .

Links

Crossrefs

Cf. A000594, A013973 (E_6), A289061, A289062.
Cf. A000521 (j), A007242, A014708, A007240.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^2 / Product[(1 - x^k)^24, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 09 2017 *)
a[ n_] := SeriesCoefficient[ q Series[ 1728 (KleinInvariantJ[Log[q] / (2 Pi I)] - 1), {q, 0, n}], {q, 0, n}]; (* Michael Somos, Mar 31 2019 *)
  • PARI
    {a(n) = my(A, U1, U2); if( n<0, 0, A = x * O(x^n); U1 = eta(x + A)^24; U2 = eta(x^2 + A)^24; polcoeff( (U1 - 512*x * U2)^2 * (U1 + 64*x * U2) / (U1^2 * U2), n))}; /* Michael Somos, Mar 31 2019 */

Formula

G.f.: Product_{k>=1} (1-q^k)^A289061(k).
a(n) = A000521(n-1) for n = 0 and n > 1.
a(n) ~ exp(4*Pi*sqrt(n)) / (sqrt(2) * n^(3/4)). - Vaclav Kotesovec, Jul 09 2017
G.f.: q * (j(q) - 1728) where j(q) is a modular function. - Michael Somos, Mar 31 2019

A299694 Coefficients in expansion of (E_4^3/E_6^2)^(1/144).

Original entry on oeis.org

1, 12, 1512, 813744, 281434656, 129501949608, 56296822560480, 26218237904433888, 12242575532254540032, 5850239653863742634172, 2820869122426120317439152, 1375631026432164061822527120, 675950202173640832786529615232
Offset: 0

Views

Author

Seiichi Manyama, Feb 16 2018

Keywords

Links

Crossrefs

(E_4^3/E_6^2)^(k/288): A289365 (k=1), this sequence (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), A289209 (k=288).
Cf. A004009 (E_4), A013973 (E_6), A296609.

Programs

  • Mathematica
    terms = 13;
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}];
(E4[x]^3/E6[x]^2)^(1/144) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Formula

Convolution inverse of A296609.
a(n) ~ 2^(1/18) * Pi^(1/24) * exp(2*Pi*n) / (3^(1/144) * Gamma(1/72) * Gamma(1/4)^(1/18) * n^(71/72)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A296609(n) ~ -sin(Pi/72) * exp(4*Pi*n) / (72*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A299696 Coefficients in expansion of (E_4^3/E_6^2)^(1/96).

Original entry on oeis.org

1, 18, 2322, 1234116, 430292646, 197681749128, 86165040337452, 40145493017336976, 18768723217958523222, 8975036477140737601806, 4331009172188712335053032, 2113419430011730408087143924, 1039122180212218474089489166980
Offset: 0

Views

Author

Seiichi Manyama, Feb 16 2018

Keywords

Links

Crossrefs

(E_4^3/E_6^2)^(k/288): A289365 (k=1), A299694 (k=2), this sequence (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), A289209 (k=288).
Cf. A004009 (E_4), A013973 (E_6), A296614.

Programs

  • Mathematica
    terms = 13;
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}];
(E4[x]^3/E6[x]^2)^(1/96) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Formula

Convolution inverse of A296614.
a(n) ~ 2^(1/12) * Pi^(1/16) * exp(2*Pi*n) / (3^(1/96) * Gamma(1/48) * Gamma(1/4)^(1/12) * n^(47/48)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A296614(n) ~ -sin(Pi/48) * exp(4*Pi*n) / (48*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A299697 Coefficients in expansion of (E_4^3/E_6^2)^(1/72).

Original entry on oeis.org

1, 24, 3168, 1663776, 584685312, 268219092816, 117214929608832, 54637244971358016, 25574598700199847936, 12238100148358426410360, 5910293921259795914011968, 2885917219371433467109558368, 1419817980186833008095972357120
Offset: 0

Views

Author

Seiichi Manyama, Feb 16 2018

Keywords

Links

Crossrefs

(E_4^3/E_6^2)^(k/288): A289365 (k=1), A299694 (k=2), A299696 (k=3), this sequence (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), A289209 (k=288).
Cf. A004009 (E_4), A013973 (E_6), A296652.

Programs

  • Mathematica
    terms = 13;
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}];
(E4[x]^3/E6[x]^2)^(1/72) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Formula

Convolution inverse of A296652.
a(n) ~ 2^(1/9) * Pi^(1/12) * exp(2*Pi*n) / (3^(1/72) * Gamma(1/36) * Gamma(1/4)^(1/9) * n^(35/36)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A296652(n) ~ -sin(Pi/36) * exp(4*Pi*n) / (36*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A299698 Coefficients in expansion of (E_4^3/E_6^2)^(1/48).

Original entry on oeis.org

1, 36, 4968, 2551824, 910405152, 416585268216, 182967944992992, 85373023607994528, 40055910812083687680, 19194979975339075406388, 9284600439037161721276848, 4539375955473797523355108272, 2236041702620444573315950439808
Offset: 0

Views

Author

Seiichi Manyama, Feb 16 2018

Keywords

Links

Crossrefs

(E_4^3/E_6^2)^(k/288): A289365 (k=1), A299694 (k=2), A299696 (k=3), A299697 (k=4), this sequence (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), A289209 (k=288).
Cf. A004009 (E_4), A013973 (E_6), A297021.

Programs

  • Mathematica
    terms = 13;
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}];
(E4[x]^3/E6[x]^2)^(1/48) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Formula

Convolution inverse of A297021.
a(n) ~ 2^(1/6) * Pi^(1/8) * exp(2*Pi*n) / (3^(1/48) * Gamma(1/24) * Gamma(1/4)^(1/6) * n^(23/24)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A297021(n) ~ -sin(Pi/24) * exp(4*Pi*n) / (24*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A299943 Coefficients in expansion of (E_4^3/E_6^2)^(1/36).

Original entry on oeis.org

1, 48, 6912, 3479616, 1259268096, 575044765344, 253777092387840, 118545813515338368, 55748828845833043968, 26753648919849657887472, 12960874757914028815661568, 6344939709971525751086888640, 3129285552537639403735326646272
Offset: 0

Views

Author

Seiichi Manyama, Feb 22 2018

Keywords

Links

Crossrefs

(E_4^3/E_6^2)^(k/288): A289365 (k=1), A299694 (k=2), A299696 (k=3), A299697 (k=4), A299698 (k=6), this sequence (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), A289209 (k=288).
Cf. A004009 (E_4), A013973 (E_6), A299422.

Programs

  • Mathematica
    terms = 13;
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}];
(E4[x]^3/E6[x]^2)^(1/36) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

Formula

Convolution inverse of A299422.
a(n) ~ c * exp(2*Pi*n) / n^(17/18), where c = 2^(2/9) * Pi^(1/6) / (3^(1/36) * Gamma(1/4)^(2/9) * Gamma(1/18)) = 0.0588537525900341685779220592527938... - Vaclav Kotesovec, Mar 04 2018
a(n) * A299422(n) ~ -sin(Pi/18) * exp(4*Pi*n) / (18*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A299949 Coefficients in expansion of (E_4^3/E_6^2)^(1/32).

Original entry on oeis.org

1, 54, 7938, 3958956, 1442594502, 658201268952, 291148964582796, 136084851675471024, 64069809910723011222, 30769281599576554087722, 14917804015099613922436392, 7307669924831130556163175612, 3606311646826590340455185471940
Offset: 0

Views

Author

Seiichi Manyama, Feb 22 2018

Keywords

Links

Crossrefs

(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), this sequence (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), A289209 (k=288).
Cf. A004009 (E_4), A013973 (E_6), A299862.

Programs

  • Mathematica
    terms = 13;
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}];
(E4[x]^3/E6[x]^2)^(1/32) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

Formula

Convolution inverse of A299862.
a(n) ~ c * exp(2*Pi*n) / n^(15/16), where c = 2^(1/4) * Pi^(3/16) / (3^(1/32) * Gamma(1/4)^(1/4) * Gamma(1/16)) = 0.06666699751397812787469360011212565... - Vaclav Kotesovec, Mar 04 2018
a(n) * A299862(n) ~ -sin(Pi/16) * exp(4*Pi*n) / (16*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A299950 Coefficients in expansion of (E_4^3/E_6^2)^(1/18).

Original entry on oeis.org

1, 96, 16128, 7622784, 2900355072, 1319081479488, 592274331915264, 278167185566287104, 131973896384325992448, 63712327450686749464032, 31055582715009234813891072, 15282363171869402875165461888, 7574187854327285047920802652160
Offset: 0

Views

Author

Seiichi Manyama, Feb 22 2018

Keywords

Links

Crossrefs

(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), this sequence (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), A289209 (k=288).
Cf. A004009 (E_4), A013973 (E_6), A299856.

Programs

  • Mathematica
    terms = 13;
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}];
(E4[x]^3/E6[x]^2)^(1/18) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

Formula

Convolution inverse of A299856.
a(n) ~ c * exp(2*Pi*n) / n^(8/9), where c = 2^(4/9) * Pi^(1/3) / (3^(1/18) * Gamma(1/4)^(4/9) * Gamma(1/9)) = 0.124111089715926449273529850774692739948955... - Vaclav Kotesovec, Mar 04 2018
a(n) * A299856(n) ~ -sin(Pi/9) * exp(4*Pi*n) / (9*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A299951 Coefficients in expansion of (E_4^3/E_6^2)^(1/16).

Original entry on oeis.org

1, 108, 18792, 8775216, 3375768096, 1535055129576, 691959629136096, 325485731190285792, 154751723387164258560, 74822912718767823810204, 36526619326785857845042608, 17998154668247683887778684176, 8931078840823632559970453020032
Offset: 0

Views

Author

Seiichi Manyama, Feb 22 2018

Keywords

Links

Crossrefs

(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), this sequence (k=18), A299953 (k=24), A299993 (k=32), A299994 (k=36), A300052 (k=48), A300053 (k=72), A300054 (k=96), A300055 (k=144), A289209 (k=288).
Cf. A004009 (E_4), A013973 (E_6), A299857.

Programs

  • Mathematica
    terms = 13;
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}];
(E4[x]^3/E6[x]^2)^(1/16) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

Formula

Convolution inverse of A299857.
a(n) ~ sqrt(2) * Pi^(3/8) * exp(2*Pi*n) / (3^(1/16) * Gamma(1/8) * sqrt(Gamma(1/4)) * n^(7/8)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A299857(n) ~ -sin(Pi/8) * exp(4*Pi*n) / (8*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018
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