A004009 -id:A004009 - cp's OEIS frontend

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.

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

Offset: 0

Seiichi Manyama, Feb 18 2017

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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

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 *)

A282474 Coefficients in q-expansion of E_4^8, where E_4 is the Eisenstein series A004009.

Original entry on oeis.org

1, 1920, 1630080, 803228160, 253366181760, 53205643249920, 7498254194403840, 699684356363412480, 42100628403784982400, 1614922125605880493440, 42332208491309728078080, 812648422343847344279040, 12060223533365891970132480

Offset: 0

Seiichi Manyama, Feb 16 2017

nonn

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

Cf. A004009 (E_4), A008410 (E_4^2), A008411 (E_4^3), A282012 (E_4^4), A282015 (E_4^5), A282330 (E_4^6), A282402 (E_4^7), this sequence (E_4^8).

terms = 13;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E4[x]^8 + 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 *)

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(%);

A377220 Expansion of (1/x) * series_reversion(x*E_4(x)), where E_4(x) denotes the Eisenstein series of weight 4 (see A004009).

Original entry on oeis.org

1, -240, 113040, -66534720, 43859560080, -30976854078240, 22919806575299520, -17536455012714130560, 13761543459443537811600, -11015192093055645841813680, 8958361831335008460574345440, -7381454927286057227098811282880, 6148958599311807793865548969813440, -5169975617288319668409172392988655520

Offset: 0

Peter Bala, Nov 07 2024

The 8th root of the power series E_4(x) has integral coefficients. See A108091. The 8th root of the g.f. of the present sequence also has integral coefficients. See A377221.

More generally if f(x) = g(x)^n, where g(x) = 1 + g_1*x + g_2*x^2 + ... is a power series with integral coefficients, then both the power series (1/x) * series_reversion(x*f(x)) and (1/x) * series_reversion(x/f(x)) are also equal to the n-th powers of integral power series.

			The 8th root of the g.f. A(x)^(1/8) = (1 - 240*x + 113040*x^2 - 66534720*x^3 + 43859560080*x^4 - 30976854078240*x^5 + 22919806575299520*x^6 +...)^(1/8) = 1 - 30*x + 10980*x^2 - 5822040*x^3 + 3623245710*x^4 - 2467207358280*x^5 + 1779938570782440*x^6 + .... lies in Z[[x]]. See A377221.

Cf. A004009, A108091, A377221, A377222, A377223.

with(numtheory):
Order := 30:
E_4 := 1 + 240*add(sigma[3](n)*x^n, n = 1..30):
solve(series(x*E_4, x) = y, x):
seq(coeftayl(series((%/y), y), y = 0, n), n = 0..20);

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]

A145312 Coefficients in expansion of E''_4(q), where E_4 is the Eisenstein series in A004009.

Original entry on oeis.org

4320, 40320, 210240, 604800, 1814400, 3467520, 7862400, 13080960, 24494400, 35164800, 64753920, 82293120, 135233280, 177811200, 269625600, 320785920, 500346720, 563068800, 838857600, 970905600, 1329229440, 1477681920, 2170022400, 2268144000, 3085992000

Offset: 0

N. J. A. Sloane, Feb 28 2009

nonn

Cf. A004009, A145346.

terms = 25;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms+1}];
E4''[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

cp's OEIS Frontend

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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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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A282474 Coefficients in q-expansion of E_4^8, where E_4 is the Eisenstein series A004009.

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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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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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A377220 Expansion of (1/x) * series_reversion(x*E_4(x)), where E_4(x) denotes the Eisenstein series of weight 4 (see A004009).

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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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A145312 Coefficients in expansion of E''_4(q), where E_4 is the Eisenstein series in A004009.

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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.

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.