A280869 -id:A280869 - cp's OEIS frontend

A126861 Coefficients in quasimodular form 12*F_3(q) of level 1 and weight 12.

0, 0, 1, 80, 1224, 9152, 45276, 170784, 534464, 1438848, 3507102, 7711600, 16053728, 30831552, 57578072, 100382304, 173117952, 280579200, 455656725, 697508496, 1079398256, 1580599552, 2351610612, 3315523424, 4785293568, 6534524160, 9173253878, 12226860576

Offset: 0

N. J. A. Sloane, Mar 15 2007

nonn

			12*F_3(q) = q^2 + 80*q^3 + 1224*q^4 + 9152*q^5 + 45276*q^6 + 170784*q^7 + 534464*q^8 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
B. Mazur, Perturbations, deformations and variations (and "near-misses") in geometry, physics, and number theory, Bull. Amer. Math. Soc., 41 (2004), 307-336.

Cf. A126858, A126859, A126860.

Cf. A280024 (E_2^4*E_4), A308285 (E_2^6), A282752 (E_2^2*E_4^2), A008411 (E_4^3), A282780 (E_2^3*E_6), A282102 (E_2*E_4*E_6), A280869 (E_6^2).

F_3(q) = (15*E(2)^4*E(4) - 6*E(2)^6 - 12*E(2)^2*E(4)^2 + 7*E(4)^3 + 4*E(2)^3*E(6) - 12*E(2)*E(4)*E(6) + 4*E(6)^2)/35831808, where E(k) is the normalized Eisenstein series of weight k (cf. A006352, etc.).

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

Original entry on oeis.org

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.

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 *)

A290010 Coefficients in expansion of 691E_6^2E_12.

Original entry on oeis.org

691, -631008, 220745952, -97839374976, 19569814631136, 4535334591238080, 303188772007707264, 10484562664742729472, 226123357637141848800, 3395301924051385082784, 38309171317486687195200, 343029821520447762390144, 2537867756328299498904192

Offset: 0

Seiichi Manyama, Jul 17 2017

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

Cf. A013973 (E_6), A029828 (691*E_12), A280869 (E_6^2).

Cf. A290009.

terms = 13;
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E12[x_] = 1 + (65520/691)*Sum[k^11*x^k/(1 - x^k), {k, 1, terms}];
691*E6[x]^2*E12[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 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

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 *)

cp's OEIS Frontend

A126861 Coefficients in quasimodular form 12*F_3(q) of level 1 and weight 12.

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A282331 Coefficients in q-expansion of E_6^4, where E_6 is the Eisenstein series 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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A290010 Coefficients in expansion of 691*E_6^2*E_12.

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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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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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A290010 Coefficients in expansion of 691E_6^2E_12.

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.