A281374 -id:A281374 - cp's OEIS frontend

A282012 Coefficients in q-expansion of E_4^4, where E_4 is the Eisenstein series shown in A004009.

1, 960, 354240, 61543680, 4858169280, 137745912960, 2120861041920, 21423820362240, 158753769048000, 928983317334720, 4512174992346240, 18847874280625920, 69518972236842240, 230951926208599680, 701949379778818560, 1975788826748167680

Offset: 0

Seiichi Manyama, Feb 04 2017

nonn

Also coefficients in q-expansion of E_8^2.

G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012, See p. 207.

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

Cf. A004009 (E_4), A008410 (E_4^2), A008411 (E_4^3), this sequence (E_4^4), A282015 (E_4^5).
Cf. A281374 (E_2^2), A008410 (E_4^2), A280869 (E_6^2), this sequence (E_8^2), A282292 (E_10^2).
Cf. A013963, A027364, A281876.

terms = 16;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E4[x]^4 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

G.f.: (1 + 240 Sum_{i>=1} i^3 q^i/(1-q^i))^4.

16320 * A013963(n) = 3617 * a(n) - 3456000 * A027364(n) for n > 0.

A282102 Coefficients in q-expansion of E_2E_4E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

1, -288, -129168, -1927296, 65152656, 1535768640, 15223408704, 98001292032, 474055120080, 1870878793824, 6312358836000, 18835985199744, 50831420617152, 126257508465984, 292348744636032, 637474437331200, 1319883180896592, 2610964045674432, 4963491913583664

Offset: 0

Seiichi Manyama, Feb 06 2017

sign

The series expansion of the 12th root of the generating function gives A341801. - Peter Bala, Feb 23 2021

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

Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A013974 (E_10).

Cf. A281374 (E_2^2), A282019 (E_2*E_4), A282096 (E_2*E_6), A282101 (E_2*E_8), this sequence (E_2*E_10), A341801.

terms = 19;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
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}];
E2[x]*E4[x]*E6[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 23 2018 *)

A282101 Coefficients in q-expansion of E_2*E_4^2, where E_2, E_4 are the Eisenstein series shown in A006352, A004009, respectively.

Original entry on oeis.org

1, 456, 50328, -470496, -21784008, -234371664, -1446514848, -6502690752, -23328111240, -71276388312, -191952331632, -468159788448, -1052750026272, -2212261706256, -4394299104576, -8303419066176, -15060718806024, -26284654025712, -44471780630856

Offset: 0

Seiichi Manyama, Feb 06 2017

sign

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

Cf. A006352 (E_2), A004009 (E_4), A008410 (E_8).

Cf. A281374 (E_2^2), A282019 (E_2*E_4), A282096 (E_2*E_6), this sequence (E_2*E_8), A282102 (E_2*E_10).

terms = 19;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E2[x]*E4[x]^2 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 23 2018 *)

A282292 Coefficients in q-expansion of E_10^2, where E_10 is the Eisenstein series A013974.

Original entry on oeis.org

1, -528, -201168, 61114944, 20946935856, 1443146395680, 46053422547264, 861726789128832, 10894843149545520, 102119072037503664, 755968133350219680, 4623420033182073024, 24151660069581371712, 110516194189880866464, 451789196756619249792

Offset: 0

Seiichi Manyama, Feb 11 2017

sign

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

Cf. A281374 (E_2^2), A008410 (E_4^2), A280869 (E_6^2), A282012 (E_8^2), this sequence (E_10^2).

terms = 15;
E10[x_] = 1 - 264*Sum[k^9*x^k/(1 - x^k), {k, 1, terms}];
E10[x]^2 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 23 2018 *)

A282096 Coefficients in q-expansion of E_2*E_6, where E_2, E_6 are the Eisenstein series shown in A006352, A013973, respectively.

Original entry on oeis.org

1, -528, -4608, 312384, 3664416, 21745440, 86782464, 276703872, 741794400, 1758969264, 3797729280, 7568097984, 14222957952, 25253852064, 43166426112, 70518360960, 112406614752, 172631876832, 260795119104, 381636168000, 552633117120, 778105665024

Offset: 0

Seiichi Manyama, Feb 06 2017

sign

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

Cf. A006352 (E_2), A013973 (E_6).

Cf. A281374 (E_2^2), A282019 (E_2*E_4), this sequence (E_2*E_6), A282101 (E_2*E_8), A282102 (E_2*E_10).

terms = 22;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E2[x]*E6[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

A282208 Coefficients in q-expansion of E_2^2*E_4, where E_2 and E_4 are respectively the Eisenstein series A006352 and A004009.

Original entry on oeis.org

1, 192, -8928, 9984, 1420896, 11433600, 53760384, 187233792, 533725920, 1327018944, 2953851840, 6060858624, 11611915392, 21030301824, 36387585792, 60357358080, 97020376032, 150755202432, 229107724704, 338493223680, 492378465600, 698632525824, 980953593984

Offset: 0

Seiichi Manyama, Feb 09 2017

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

Cf. A006352 (E_2), A004009 (E_4), A281374 (E_2^2), A282019 (E_2*E_4), A008410 (E_4^2 = E_8), A282018 (E_2^3), this sequence (E_2^2*E_4), A282101 (E_2*E_4^2), A008411 (E_4^3).

terms = 23;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E2[x]^2*E4[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

A282210 Coefficients in q-expansion of E_2^4, where E_2 is the Eisenstein series shown in A006352.

Original entry on oeis.org

1, -96, 3168, -34944, -107808, 1955520, 16829568, 76708608, 258593760, 715480608, 1729546560, 3771497088, 7581237888, 14296261056, 25520442624, 43590539520, 71582414304, 113752634688, 175604039136, 264097115520, 388619703360, 559658001408, 792716685696

Offset: 0

Seiichi Manyama, Feb 09 2017

sign

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

Cf. A006352 (E_2), A281374 (E_2^2), A282018 (E_2^3), this sequence (E_2^4).

terms = 23;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E2[x]^4 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A377973 Expansion of the 96th root of the series 2*E_2(x) - E_2(x)^2, where E_2 is the Eisenstein series of weight 2.

Original entry on oeis.org

1, 0, -6, -36, -1812, -20748, -773340, -12237456, -386587650, -7368446268, -211914644940, -4517757977820, -123221458979940, -2814502962357420, -74551748141034552, -1778129476480366320, -46377354051910716180, -1137191336376638407704, -29438532048777299115090, -735051729258136807204140

Offset: 0

Peter Bala, Nov 13 2024

Let R = 1 + x*Z[[x]] denote the set of integer power series with constant term equal to 1. Let P(n) = {g^n, g in R}. The Eisenstein series E_2(x) lies in P(4) (Heninger et al.). Hence E_2(x)^2 lies in P(8).
We claim that the series 2*E_2(x) - E_2(x)^2 belongs to P(96).
Proof.
E_2(x) = 1 - 24*Sum_{n >= 1} sigma_1(n)*x^n.
Hence,
2*E_2(x) - E_2(x)^2 = 1 - (24^2)*(Sum_{n >= 1} sigma_1(n)*x^n )^2 is in the set R.
Hence, 2*E_2(x) - E_2(x)^2 == 1 (mod 24^2) == 1 (mod (2^6)*(3^2)).
It follows from Heninger et al., Theorem 1, Corollary 2, that the series 2*E_2(x) - E_2(x)^2 belongs to P((2^5)*3) = P(96). End Proof.

N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Cf. A006352 (E_2), A281374 (E_2)^2, A289392 ((E_2)^(1/4)), A341801, A341871 - A341875, A377974, A377975, A377976, A377977.

with(numtheory):
E := proc (k) local n, t1; t1 := 1 - 2*k*add(sigma[k-1](n)*q^n, n = 1..30)/bernoulli(k); series(t1, q, 30) end:
seq(coeftayl((2*E(2) - E(2)^2)^(1/96), q = 0, n),n = 0..20);

terms = 20; E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}]; CoefficientList[Series[(2*E2[x] - E2[x]^2)^(1/96), {x, 0, terms}], x] (* Vaclav Kotesovec, Aug 03 2025 *)

a(n) ~ c / (r^n * n^(97/96)), where r = A211342 = 0.03727681029645165815098... and c = -0.0104397599261506010365791466642760245638473040812140699981294533624... - Vaclav Kotesovec, Aug 03 2025

A282431 Coefficients in q-expansion of E_2^5, where E_2 is the Eisenstein series A006352.

Original entry on oeis.org

1, -120, 5400, -104160, 511800, 6770736, -19504800, -452207040, -2959622280, -12932941080, -44497080432, -129918587040, -335811977760, -788655411600, -1714912983360, -3498061536576, -6761506680840, -12481939678320, -22138262633160, -37922739116640

Offset: 0

Seiichi Manyama, Feb 15 2017

sign

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

Cf. this sequence (E_2^5), A282015 (E_4^5), A282433 (E_6^5).

Cf. A006352 (E_2), A281374 (E_2^2), A282018 (E_2^3), A282210 (E_2^4), this sequence (E_2^5).

terms = 20;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E2[x]^5 + 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

cp's OEIS Frontend

A282012 Coefficients in q-expansion of E_4^4, where E_4 is the Eisenstein series shown in A004009.

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A282102 Coefficients in q-expansion of E_2*E_4*E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

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A282101 Coefficients in q-expansion of E_2*E_4^2, where E_2, E_4 are the Eisenstein series shown in A006352, A004009, respectively.

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A282292 Coefficients in q-expansion of E_10^2, where E_10 is the Eisenstein series A013974.

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A282096 Coefficients in q-expansion of E_2*E_6, where E_2, E_6 are the Eisenstein series shown in A006352, A013973, respectively.

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A282208 Coefficients in q-expansion of E_2^2*E_4, where E_2 and E_4 are respectively the Eisenstein series A006352 and A004009.

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A282210 Coefficients in q-expansion of E_2^4, where E_2 is the Eisenstein series shown in A006352.

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A377973 Expansion of the 96th root of the series 2*E_2(x) - E_2(x)^2, where E_2 is the Eisenstein series of weight 2.

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A282431 Coefficients in q-expansion of E_2^5, where E_2 is the Eisenstein series A006352.

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A282102 Coefficients in q-expansion of E_2E_4E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

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.