A013973 -id:A013973 - cp's OEIS frontend

A282331 Coefficients in q-expansion of E_6^4, where E_6 is the Eisenstein series A013973.

1, -2016, 1457568, -411997824, 16227967392, 6497071680960, 440015323483008, 15172068869975808, 327221898778968480, 4913597307075535008, 55440561879404210880, 496424806634688962688, 3672744471642078903168, 23148319448757751932096

Offset: 0

Seiichi Manyama, Feb 12 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Eisenstein Series.

Cf. A013973 (E_6), A280869 (E_6^2), A282253 (E_6^3), this sequence (E_6^4).

Cf. A282210 (E_2^4), A282012 (E_4^4), this sequence (E_6^4).

terms = 14;
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E6[x]^4 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

E6(q)^4 = (1 - 504 Sum_{i>=1} sigma_5(i)q^i)^4 where sigma_5(n) is A001160.

A282357 Coefficients in q-expansion of E_4^2*E_6^3, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

Original entry on oeis.org

1, -1032, 48312, 171162336, -6444771144, -10105554483504, -1037089473751584, -48959817978105408, -1378102838778701640, -26186640301645703016, -364779940958775418032, -3952291567255306906464, -34798629548716507265568, -257403564989318828310384

Offset: 0

Seiichi Manyama, Feb 13 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A008410 (E_4^2 = E_8), A058550 (E_4^2*E_6 = E_14), A282292 (E_4^2*E_6^2 = E_10^2), this sequence (E_4^2*E_6^3).

terms = 14;
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]^2*E6[x]^3 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A282382 Coefficients in q-expansion of E_4^6*E_6, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

Original entry on oeis.org

1, 936, 134568, -173988576, -104617833048, -27210540914064, -3910401774129888, -322823174243838912, -15429983442476298840, -469709326015243815672, -9973673112569954220432, -158215072218253260221088, -1972939697011615168926432

Offset: 0

Seiichi Manyama, Feb 16 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A004009 (E_4), A013973 (E_6), A013974 (E_4*E_6 = E_10), A058550 (E_4^2*E_6 = E_14), A282000 (E_4^3*E_6), A282047 (E_4^4*E_6), A282048 (E_4^5*E_6), this sequence (E_4^6*E_6).

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]^6*E6[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A282403 Coefficients in q-expansion of E_4^4*E_6^2, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

Original entry on oeis.org

1, -48, -392688, -67089216, 37279185936, 15066490704480, 2098369148842944, 134803101024250752, 4960096515113176080, 119289357755096403984, 2051412780505054295520, 26894040676649639982144, 281804014682888704101312

Offset: 0

Seiichi Manyama, Feb 14 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A280869 (E_6^2), A282287 (E_4*E_6^2), A282292 (E_4^2*E_6^2 = E_10^2), A282332 (E_4^3*E_6^2), this sequence (E_4^4*E_6^2).

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]^4* E6[x]^2 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A282433 Coefficients in q-expansion of E_6^5, where E_6 is the Eisenstein series A013973.

Original entry on oeis.org

1, -2520, 2457000, -1113204960, 199879986600, 4992350445936, -3054519828108000, -316433406335739840, -15444821445342229080, -469944493113793897080, -9973874479528786860432, -158211337782226162119840, -1972932224893221543809760

Offset: 0

Seiichi Manyama, Feb 15 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A282431 (E_2^5), A282015 (E_4^5), this sequence (E_6^5).

Cf. A013973 (E_6), A280869 (E_6^2), A282253 (E_6^3), A282331 (E_6^4), this sequence (E_6^5).

terms = 13;
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E6[x]^5 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A282461 Coefficients in q-expansion of E_4^3*E_6^3, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

Original entry on oeis.org

1, -792, -197208, 180534816, 34731625896, -11282282306064, -3475192229286624, -319729598062193088, -15436589476561121880, -469831003553540798136, -9973761497118317484432, -158213220814147434639264, -1972935965978751882433248

Offset: 0

Seiichi Manyama, Feb 16 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A013974 (E_4*E_6 = E_10), A282292 (E_4^2*E_6^2 = E_10^2), this sequence (E_4^3*E_6^3 = E_10^3).

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]^3 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A319134 Expansion of -((25E_4^4 - 49E_6^2E4) + 48E_6E_4^2E_2 + (-49E_4^3 + 25E_6^2)E_2^2)/(3657830400delta^2) where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively and delta is A000594.

Original entry on oeis.org

1, 86, 3750, 109672, 2419462, 43021728, 643548464, 8343640624, 95835049605, 991606081332, 9364586280842, 81571540591968, 661034448807902, 5019357866562208, 35927279225314344, 243657157464337888, 1572638456431119570, 9696997279843999470, 57313953586222481126, 325672739267123628976

Offset: 1

Seiichi Manyama, Sep 11 2018

nonn

			((25*E_4^4 - 49*E_6^2*E4) + 48*E_6*E_4^2*E_2 + (-49*E_4^3 + 25*E_6^2)*E_2^2)/(delta^2) =  - 3657830400*q - 314573414400*q^2 - 13716864000000*q^3 - 401161575628800*q^4 - ... .

Seiichi Manyama, Table of n, a(n) for n = 1..5000
H. Cohn, A. Kumar, S. Miller, D. Radchenko, M. Viazovska, The sphere packing problem in dimension 24, Annals of Mathematics, 185 (3) (2017), 1017-1033.
Wikipedia, Sphere packing

Cf. A000594, A006352 (E_2), A004009 (E_4), A013973 (E_6), A082558, A281373,

About the numerator: A282012 (E_4^4), A282287 (E_6^2*E_4), A282596 (E_6*E_4^2*E_2), A008411 (E_4^3), A280869 (E_6^2), A281374 (E_2^2).

nmax = 25; E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, nmax + 1}] + O[x]^(nmax + 1); E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, nmax + 1}] + O[x]^(nmax + 1); E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, nmax + 1}] + O[x]^(nmax + 1); Rest[CoefficientList[Series[-((25*E4[x]^4 - 49*E6[x]^2*E4[x]) + 48*E6[x]*E4[x]^2*E2[x] + (-49*E4[x]^3 + 25*E6[x]^2)* E2[x]^2) / (3657830400 * x^2 * QPochhammer[x]^48), {x, 0, nmax}], x]] (* Vaclav Kotesovec, Sep 12 2018 *)

a(n) ~ exp(4*Pi*sqrt(2*n)) / (132300 * 2^(1/4) * Pi^2 * n^(23/4)). - Vaclav Kotesovec, Sep 12 2018

A282404 Coefficients in q-expansion of E_4*E_6^4, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

Original entry on oeis.org

1, -1776, 975888, -66529344, -79516693488, 9511628122080, 2031621786790848, 134911299030780288, 4962883791154433040, 119289719378991436368, 2051366007318600561120, 26893975935849646148928, 281804567385216854182848

Offset: 0

Seiichi Manyama, Feb 14 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A013974 (E_4*E_6 = E_10), A282287 (E_4*E_6^2), A282328 (E_4*E_6^3), this sequence (E_4*E_6^4).

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]*E6[x]^4 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A282541 Coefficients in q-expansion of E_4^5*E_6^2, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

Original entry on oeis.org

1, 192, -402048, -161431296, 20329262976, 23865942948480, 5794392238723584, 671204645516954112, 41947216018774335360, 1615253348424607402944, 42337765240473386384640, 812656088633074046171904, 12060155362281020231526912

Offset: 0

Seiichi Manyama, Feb 17 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A280869 (E_6^2), A282287 (E_4*E_6^2), A282292 (E_4^2*E_6^2 = E_10^2), A282332 (E_4^3*E_6^2), A282403 (E_4^4*E_6^2), this sequence (E_4^5*E_6^2).

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]^5* E6[x]^2 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A282543 Coefficients in q-expansion of E_4^2*E_6^4, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

Original entry on oeis.org

1, -1536, 551808, 163854336, -93387735168, -9709554816000, 4142226444876288, 642510156233453568, 41792421673548259200, 1615606968766288470528, 42343208407470359036160, 812663841518551604717568, 12060089370317565140003328

Offset: 0

Seiichi Manyama, Feb 17 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A008410 (E_4^2 = E_8), A058550 (E_4^2*E_6 = E_14), A282292 (E_4^2*E_6^2 = E_10^2), A282357 (E_4^2*E_6^3), this sequence (E_4^2*E_6^4).

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]^2*E6[x]^4 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

cp's OEIS Frontend

A282331 Coefficients in q-expansion of E_6^4, where E_6 is the Eisenstein series A013973.

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A282357 Coefficients in q-expansion of E_4^2*E_6^3, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

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A282382 Coefficients in q-expansion of E_4^6*E_6, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

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A282403 Coefficients in q-expansion of E_4^4*E_6^2, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

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A282433 Coefficients in q-expansion of E_6^5, where E_6 is the Eisenstein series A013973.

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A282461 Coefficients in q-expansion of E_4^3*E_6^3, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

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A319134 Expansion of -((25*E_4^4 - 49*E_6^2*E4) + 48*E_6*E_4^2*E_2 + (-49*E_4^3 + 25*E_6^2)*E_2^2)/(3657830400*delta^2) where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively and delta is A000594.

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A282404 Coefficients in q-expansion of E_4*E_6^4, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

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A282541 Coefficients in q-expansion of E_4^5*E_6^2, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

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A282543 Coefficients in q-expansion of E_4^2*E_6^4, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

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A319134 Expansion of -((25E_4^4 - 49E_6^2E4) + 48E_6E_4^2E_2 + (-49E_4^3 + 25E_6^2)E_2^2)/(3657830400delta^2) where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively and delta is A000594.