A006352 -id:A006352 - cp's OEIS frontend

A280024 Coefficients in q-expansion of E_2^4*E_4, where E_2 and E_4 are respectively the Eisenstein series A006352 and A004009.

1, 144, -17712, 524736, -2279088, -79760160, 71126208, 7093116288, 65399933520, 370698709968, 1592500629600, 5659924638528, 17465468914368, 48233085519456, 121766302456704, 285303917520000, 627654170451024, 1308136029869088, 2601247015228176

Offset: 0

Seiichi Manyama, Feb 22 2017

sign

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

Cf. A282019 (E_2*E_4), A282208 (E_2^2*E_4), A282586 (E_2^3*E_4), this sequence (E_2^4*E_4).

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]^4*E4[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A281371 Coefficients in q-expansion of (E_2*E_4 - E_6)^2/518400, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

0, 0, 1, 36, 492, 3608, 18828, 74760, 250352, 717984, 1866558, 4365580, 9635472, 19639032, 38559416, 71222616, 128258496, 219619968, 370366101, 597550068, 955638824, 1471571136, 2253173892, 3335433368, 4932972864, 7064391840, 10133162774, 14128072488, 19743952032, 26864847352

Offset: 0

N. J. A. Sloane, Feb 04 2017

nonn

This is (up to a constant factor), the numerator of the expression phi defined in Cohn (2017) (see phi on page 114 of the Notices version).

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Henry Cohn, A conceptual breakthrough in sphere packing, arXiv preprint arXiv:1611.01685 [math.MG], 2016; also Notices Amer. Math. Soc., 64:2 (2017), pp. 102-115.

Cf. A006352, A004009, A013973, A145094, A281372 (the square root).

with(numtheory); M:=100;
E := proc(k) local n, t1; global M;
t1 := 1-(2*k/bernoulli(k))*add(sigma[k-1](n)*q^n, n=1..M+1);
series(t1, q, M+1); end;
e2:=E(2); e4:=E(4); e6:=E(6);
t1:=series((e2*e4-e6)^2/518400,q,M+1);
seriestolist(t1);

terms = 30;
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])^2/518400 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A281373 Coefficients in q-expansion of (E_2E_4 - E_6)^2/(300(E_6^2-E_4^3)), where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

0, 1, 60, 1680, 30280, 405678, 4369680, 39729200, 315045840, 2230260741, 14340456648, 84870112272, 467160257760, 2411818867430, 11759239565472, 54457051387536, 240692336520352, 1019498573990610, 4152992658207660, 16319887656747248, 62032458633713904, 228608370781579488

Offset: 0

N. J. A. Sloane, Feb 04 2017

nonn

This is (up to a constant factor), the function phi defined in Cohn (2017) (see phi on page 114 of the Notices version).

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)
Henry Cohn, A conceptual breakthrough in sphere packing, arXiv preprint arXiv:1611.01685 [math.MG], 2016.
Henry Cohn, A conceptual breakthrough in sphere packing, Notices Amer. Math. Soc., 64:2 (2017), pp. 102-115.

Cf. A006352, A004009, A013973, A145094, A281371 (the numerator), A000594 (the denominator), A319134, A319294.

with(numtheory); M:=100;
E := proc(k) local n, t1; global M;
t1 := 1-(2*k/bernoulli(k))*add(sigma[k-1](n)*q^n, n=1..M+1);
series(t1, q, M+1); end;
e2:=E(2); e4:=E(4); e6:=E(6);
t1:=series((e2*e4-e6)^2/518400,q,M+1);
t2:=series((e4^3-e6^2)/1728,q,M+1);
t3:=series(t1/t2,q,M+1);
seriestolist(t3);

terms = 22;
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])^2/(300*(E6[x]^2 - E4[x]^3)) + O[x]^terms // CoefficientList[#, x]& // Abs (* Jean-François Alcover, Feb 27 2018 *)

a(n) ~ exp(4*Pi*sqrt(n)) / (14400 * sqrt(2) * Pi^2 * n^(7/4)). - Vaclav Kotesovec, Jun 06 2018

A282546 Coefficients in q-expansion of E_2*E_4^4, where E_2 and E_4 are respectively the Eisenstein series A006352 and A004009.

Original entry on oeis.org

1, 936, 331128, 52972704, 3355523352, 16684536816, -1540796901408, -39871325253312, -522168659242920, -4651083548616312, -31647933913392432, -175516717881381408, -827283695234707872, -3413277291552455376, -12598120840018061376, -42296015537631706176

Offset: 0

Seiichi Manyama, Feb 18 2017

sign

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

Cf. A006352 (E_2), A004009 (E_4), A282012 (E_4^4).

Cf. A282019 (E_2*E_4), A282101 (E_2*E_4^2), A282549 (E_2*E_4^3), this sequence (E_2*E_4^4).

terms = 16;
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]^4 + 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

A308285 Coefficients in q-expansion of E_2^6, where E_2 is the Eisenstein series A006352.

Original entry on oeis.org

1, -144, 8208, -225216, 2634192, 1488672, -209742912, -503961984, 8575185744, 91347182640, 524570699232, 2230073940672, 7794083954880, 23627036677536, 64145226215808, 159373702203264, 368012313906768, 798872890993632, 1644874069475664, 3234829827767616

Offset: 0

Seiichi Manyama, May 18 2019

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

E_2^b: A006352 (b=1), A281374 (b=2), A282018 (b=3), A282210 (b=4), A282431 (b=5), this sequence (b=6).

A341801 Coefficients of the series whose 12th power equals E_2E_4E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973.

Original entry on oeis.org

1, -24, -13932, -3585216, -1580941068, -628142318640, -281617154080704, -126114490533924480, -58596395743623957084, -27537281150571923942424, -13153668428658997172513880, -6345860505664230715931502912, -3091029995619009106117946403456

Offset: 0

Peter Bala, Feb 20 2021

The g.f. is the 12th root of the g.f. of A282102.

It is easy to see that E_2(x)*E_4(x)*E_6(x) == 1 - 24*Sum_{k >= 1} (k - 10*k^3 + 21*k*5)*x^k/(1 - x^k) (mod 72), and also that the integer k - 10*k^3 + 21*k*5 = k*(3*k^2 - 1)*(7^k^2 - 1) is always divisible by 3. Hence, E_2(x)*E_4(x)*E_6(x) == 1 (mod 72). It follows from Heninger et al., p. 3, Corollary 2, that the series expansion of (E_2(x)*E_4(x)* E_6(x))^(1/12) = 1 - 24*x - 13932*x^2 - 3585216*x^3 - 1580941068*x^4 - ... has integer coefficients.

N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
Wikipedia, Eisenstein series

Cf. A004009, A006352, A013973, A108091, A109817, A282102, A289392.

E(2,x) := 1 -  24*add(k*x^k/(1-x^k),   k = 1..20):
E(4,x) := 1 + 240*add(k^3*x^k/(1-x^k), k = 1..20):
E(6,x) := 1 - 504*add(k^5*x^k/(1-x^k), k = 1..20):
with(gfun): series((E(2,x)*E(4,x)*E(6,x))^(1/12), x, 20):
seriestolist(%);

A341874 Coefficients of the series whose 24th power equals E_2(x)^7/E_14(x), where E_2(x) and E_14(x) are the Eisenstein series A006352 and A058550.

Original entry on oeis.org

1, -6, 8118, 1740636, 937783902, 364856395608, 172736345164500, 78278100914583312, 37268001893898954198, 17773741638825114790854, 8624927270409695050736952, 4214914849580580859932456300, 2078204723099375850950863499028

Offset: 0

Peter Bala, Feb 23 2021

It is easy to see that E_2(x)^7/E_14(x) == 1 - 24*Sum_{k >= 1} (7*k - 11*k^13)*x^k/(1 - x^k) (mod 144), and also that the integer 7*k - k^13 is always divisible by 6. Hence, E_2(x)^7/E_14(x) == 1 (mod 144). It follows from Heninger et al., p. 3, Corollary 2, that the series expansion of (E_2(x)^7/E_14(x))^(1/24) = 1 - 6*x + 8118*x^2 + 1740636*x^3 + 937783902*x^4 + ... has integer coefficients.

N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
Wikipedia, Eisenstein series

Cf. A006352, A058550, A341871 - A341875.

E(2,x)  := 1 -  24*add(k*x^k/(1-x^k),   k = 1..20):
E(14,x) := 1 - 24*add(k^13*x^k/(1-x^k), k = 1..20):
with(gfun): series((E(2,x)^7/E(14,x))^(1/24), x, 20):
seriestolist(%);

A386813 Coefficients in q-expansion of E_2^3 * E_4^2, where E_2 and E_4 are respectively the Eisenstein series A006352 and A004009.

Original entry on oeis.org

1, 408, 28872, -2685984, 24039336, 776610576, -657274464, -112765274688, -1315204139160, -9184174537416, -47705529895632, -201727238619744, -730623451715808, -2340991131399984, -6787572064867008, -18105120840067776, -44991518932447512, -105189400371536208, -233200610257765464

Offset: 0

Vaclav Kotesovec, Aug 03 2025

sign

Cf. A006352, A004009, A386781, A386785.

terms = 20;
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}];
CoefficientList[Series[E2[x]^3*E4[x]^2, {x, 0, terms}], x]

A282020 Coefficients in q-expansion of (E_2^3 - E_2*E_4)/288, where E_2 and E_4 are the Eisenstein series shown in A006352 and A004009, respectively.

Original entry on oeis.org

0, -1, 18, 204, 788, 2250, 4968, 9688, 17640, 27747, 45900, 64548, 98448, 128674, 188496, 232200, 326864, 386478, 537354, 608380, 819000, 926688, 1214136, 1323144, 1758240, 1852625, 2401308, 2584440, 3252256, 3385170, 4374000, 4433248, 5604768, 5840208, 7143876, 7232400, 9239364, 9058858

Offset: 0

N. J. A. Sloane, Feb 06 2017

sign

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

Cf. A282018 (E_2^3), A282019 (E_2*E_4).

with(numtheory); M:=100;
E := proc(k) local n, t1; global M;
t1 := 1-(2*k/bernoulli(k))*add(sigma[k-1](n)*q^n, n=1..M+1);
series(t1, q, M+1); end;
e2:=E(2); e4:=E(4); e6:=E(6);
series((e2^3-e2*e4)/288,q,M+1);
seriestolist(%);

terms = 38;
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 + 1}];
(E2[x]^3 - E2[x]*E4[x])/288 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

a(n) = (A282018(n) - A282019(n))/288. - Seiichi Manyama, Feb 06 2017

cp's OEIS Frontend

A280024 Coefficients in q-expansion of E_2^4*E_4, where E_2 and E_4 are respectively the Eisenstein series A006352 and A004009.

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A281371 Coefficients in q-expansion of (E_2*E_4 - E_6)^2/518400, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

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A281373 Coefficients in q-expansion of (E_2*E_4 - E_6)^2/(300*(E_6^2-E_4^3)), where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

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A282546 Coefficients in q-expansion of E_2*E_4^4, where E_2 and E_4 are respectively the Eisenstein series A006352 and A004009.

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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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A308285 Coefficients in q-expansion of E_2^6, where E_2 is the Eisenstein series A006352.

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A341801 Coefficients of the series whose 12th power equals E_2*E_4*E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973.

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A341874 Coefficients of the series whose 24th power equals E_2(x)^7/E_14(x), where E_2(x) and E_14(x) are the Eisenstein series A006352 and A058550.

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A386813 Coefficients in q-expansion of E_2^3 * E_4^2, where E_2 and E_4 are respectively the Eisenstein series A006352 and A004009.

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A282020 Coefficients in q-expansion of (E_2^3 - E_2*E_4)/288, where E_2 and E_4 are the Eisenstein series shown in A006352 and A004009, respectively.

A281373 Coefficients in q-expansion of (E_2E_4 - E_6)^2/(300(E_6^2-E_4^3)), 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.

A341801 Coefficients of the series whose 12th power equals E_2E_4E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973.