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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A300055 Coefficients in expansion of (E_4^3/E_6^2)^(1/2).

Original entry on oeis.org

1, 864, 476928, 254399616, 136313874432, 72985679394624, 39084426149704704, 20929208813297429760, 11207444175842517172224, 6001488285356611750823136, 3213747681163891383409648128, 1720934927015053152217599326592
Offset: 0

Views

Author

Seiichi Manyama, Feb 23 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), A299951 (k=18), A299953 (k=24), A299993 (k=32), A299994 (k=36), A300052 (k=48), A300053 (k=72), A300054 (k=96), this sequence (k=144), A289209 (k=288).
Cf. A004009 (E_4), A013973 (E_6), A299413.

Programs

  • Mathematica
    terms = 12;
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)^(1/2) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

Formula

Convolution inverse of A299413.
a(n) ~ 16 * Pi^3 * exp(2*Pi*n) / (sqrt(3) * Gamma(1/4)^4). - Vaclav Kotesovec, Mar 04 2018

A029830 Eisenstein series E_20(q) (alternate convention E_10(q)), multiplied by 174611.

Original entry on oeis.org

174611, 13200, 6920614800, 15341851377600, 3628395292275600, 251770019531263200, 8043563916910526400, 150465416446925500800, 1902324110996589786000, 17831242688625346952400, 132000251770026451864800, 807299993919072011054400, 4217144038884527916580800, 19297347832955888660949600
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • J.-P. Serre, Course in Arithmetic, Chap. VII, Section 4.

Links

Crossrefs

Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A008410 (E_8), A013974 (E_10), A029828 (691*E_12), A058550 (E_14), A029829 (3617*E_16), A279892 (43867*E_18), this sequence (174611*E_20), A279893 (77683*E_22), A029831 (236364091*E_24).
Cf. A282015 (E_4^5), A282292 (E_4^2*E_6^2 = E_10^2).

Programs

  • Mathematica
    terms = 14;
E20[x_] = 174611 + 13200*Sum[k^19*x^k/(1 - x^k), {k, 1, terms}];
E20[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)
  • PARI
    a(n)=if(n<1,174611*(n==0),13200*sigma(n,19))

Formula

a(n) = 53361*A282015(n) + 121250*A282292(n). - Seiichi Manyama, Feb 11 2017

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

Views

Author

Seiichi Manyama, Jun 17 2017

Keywords

Examples

			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)

References

  • 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.

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

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(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

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

Views

Author

Seiichi Manyama, Jul 02 2017

Keywords

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)

Formula

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

A008704 Theta series of Niemeier lattice of type A_1^24.

Original entry on oeis.org

1, 48, 195408, 16785216, 397963344, 4629612960, 34417365696, 187489131648, 814883829840, 2975546033136, 9486545735520, 27052971581376, 70486219194048, 169931067595296, 384163605641088
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 407.

Links

Crossrefs

Cf. A004009, A013973.
Cf. A008688 - A008703.

Programs

  • Mathematica
    terms = 15; E4[q_] := 1 + 240 Sum[DivisorSigma[3, n]*q^(2 n), {n, 1, terms}]; E6[q_] := 1 - 504 Sum[DivisorSigma[5, n]*q^(2 n), {n, 1, terms}]; s = 11/18 E4[q]^3 + 7/18 E6[q]^2 + O[q]^(3 terms); Partition[ CoefficientList[s, q], 2][[All, 1]][[1 ;; terms]] (* Jean-François Alcover, Jul 06 2017 *)

Formula

This series is the q-expansion of (11*E_4(z)^3 + 7*E_6(z)^2)/18. - Daniel D. Briggs, Nov 26 2011

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

Views

Author

Seiichi Manyama, Jul 02 2017

Keywords

Examples

			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)

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)

Formula

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

Views

Author

Seiichi Manyama, Jul 02 2017

Keywords

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)

Formula

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

A289328 Coefficients in expansion of E_6^(5/12).

Original entry on oeis.org

1, -210, -37800, -10300080, -3534651750, -1351633962672, -551776752641520, -235367241169341120, -103623939263346377400, -46723958347194591810690, -21464711387762586693907248, -10009787904868201520473221840
Offset: 0

Views

Author

Seiichi Manyama, Jul 02 2017

Keywords

Crossrefs

E_6^(k/12): A109817 (k=1), A289325 (k=2), A289326 (k=3), A289327 (k=4), this sequence (k=5), A289293 (k=6).
Cf. A013973 (E_6), A288851.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^(5/12), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(5*A288851(n)/12).
a(n) ~ c * exp(2*Pi*n) / n^(17/12), where c = -5 * Gamma(1/12) * Gamma(1/4)^(20/3) / (128 * 2^(11/12) * 3^(2/3) * Pi^5 * Gamma(1/3)^2) = -0.2792181117471536554156263079143941137076647484619917046386429000631... - Vaclav Kotesovec, Jul 08 2017, updated Mar 05 2018

A289345 Coefficients in expansion of E_6^(7/12).

Original entry on oeis.org

1, -294, -40572, -9456216, -3013531458, -1095736644072, -430427492908056, -177966281438573376, -76323096421188881292, -33643171872410204427918, -15150435131179232328586968, -6940567145625149028384495432
Offset: 0

Views

Author

Seiichi Manyama, Jul 03 2017

Keywords

Crossrefs

E_6^(k/12): A109817 (k=1), A289325 (k=2), A289326 (k=3), A289327 (k=4), A289328 (k=5), A289293 (k=6), this sequence (k=7), A289346 (k=8), A289347 (k=9), A289348 (k=10), A289349 (k=11).
Cf. A013973 (E_6), A288851.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^(7/12), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(7*A288851(n)/12).
a(n) ~ c * exp(2*Pi*n) / n^(19/12), where c = -7 * Gamma(1/12) * Gamma(1/4)^(22/3) / (1024 * 6^(1/12) * Pi^7) = -0.2836006135316422535659652380776952016594933981... - Vaclav Kotesovec, Jul 08 2017, updated Mar 05 2018

A289346 Coefficients in expansion of E_6^(2/3).

Original entry on oeis.org

1, -336, -39312, -8266944, -2529479568, -895678457184, -344891780549568, -140330667583849344, -59379605532142099344, -25873741825665005773200, -11534062764689844375098592, -5236325710480558290644292672
Offset: 0

Views

Author

Seiichi Manyama, Jul 03 2017

Keywords

Crossrefs

E_6^(k/12): A109817 (k=1), A289325 (k=2), A289326 (k=3), A289327 (k=4), A289328 (k=5), A289293 (k=6), A289345 (k=7), this sequence (k=8), A289347 (k=9), A289348 (k=10), A289349 (k=11).
Cf. A013973 (E_6), A288851.

Programs

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

Formula

G.f.: Product_{n>=1} (1-q^n)^(2*A288851(n)/3).
a(n) ~ c * exp(2*Pi*n) / n^(5/3), where c = -3^(1/3) * Gamma(1/4)^(32/3) / (128 * 2^(2/3) * Pi^8 * Gamma(1/3)) = -0.258650618394676269905172499217587002338... - Vaclav Kotesovec, Jul 08 2017, updated Mar 05 2018
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