A282287 -id:A282287 - cp's OEIS frontend

A029829 Eisenstein series E_16(q) (alternate convention E_8(q)), multiplied by 3617.

3617, 16320, 534790080, 234174178560, 17524001357760, 498046875016320, 7673653657232640, 77480203842286080, 574226476491096000, 3360143509958850240, 16320498047409790080, 68172690124863440640

Offset: 0

N. J. A. Sloane

nonn

N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see p. 111.
J.-P. Serre, Course in Arithmetic, Chap. VII, Section 4.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for sequences related to Eisenstein series

Cf. A058552.

Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A008410 (E_8), A013974 (E_10), A029828 (E_12), A058550 (E_14), A029829 (E_16), A029830 (E_20), A029831 (E_24).

E := proc(k) local n,t1; t1 := 1-(2*k/bernoulli(k))*add(sigma[k-1](n)*q^n,n=1..60); series(t1,q,60); end; E(16);

terms = 12;
E16[x_] = 3617 + 16320*Sum[k^15*x^k/(1 - x^k), {k, 1, terms}];
E16[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

a(n)=if(n<1,3617*(n==0),16320*sigma(n,15))

a(n) = 1617*A282012(n) + 2000*A282287(n). - Seiichi Manyama, Feb 11 2017

A386787 a(n) = n^4*sigma_7(n).

Original entry on oeis.org

0, 1, 2064, 177228, 4227328, 48828750, 365798592, 1977329144, 8657571840, 31395415077, 100782540000, 285311685252, 749200886784, 1792160422598, 4081207353216, 8653821705000, 17730707193856, 34271896391154, 64800136718928, 116490259028540, 206415142080000, 350438089532832

Offset: 0

Vaclav Kotesovec, Aug 02 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..7500

Cf. A280022, A386783, A280025, A386784, A386785, A386786, A386788.
Cf. A013955, A282060, A282753, A386781.
Cf. A386815, A386816, A282012, A386817, A282596, A386818, A282287.

[0] cat [n^4*DivisorSigma(7, n): n in [1..35]]; // Vincenzo Librandi, Aug 03 2025

Table[n^4*DivisorSigma[7, n], {n, 0, 30}]
(* or *)
nmax = 30; CoefficientList[Series[Sum[k^4*x^k*(1 + 2036*x^k + 152637*x^(2*k) + 2203488*x^(3*k) + 9738114*x^(4*k) + 15724248*x^(5*k) + 9738114*x^(6*k) + 2203488*x^(7*k) + 152637*x^(8*k) + 2036*x^(9*k) + x^(10*k))/(1 - x^k)^12, {k, 1, nmax}], {x, 0, nmax}], x]
(* or *)
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}]; CoefficientList[Series[(33*E2[x]^4*E4[x]^2 + 110*E2[x]^2*E4[x]^3 + 13*E4[x]^4 - 132*E2[x]^3*E4[x]*E6[x] - 132*E2[x]*E4[x]^2*E6[x] + 88*E2[x]^2*E6[x]^2 + 20*E4[x]*E6[x]^2)/41472, {x, 0, terms}], x]

G.f.: Sum_{k>=1} k^4*x^k*(1 + 2036*x^k + 152637*x^(2*k) + 2203488*x^(3*k) + 9738114*x^(4*k) + 15724248*x^(5*k) + 9738114*x^(6*k) + 2203488*x^(7*k) + 152637*x^(8*k) + 2036*x^(9*k) + x^(10*k))/(1 - x^k)^12.
a(n) = (33*A386815(n) + 110*A386816(n) + 13*A282012(n) - 132*A386817(n) - 132*A282596(n) + 88*A386818(n) + 20*A282287(n))/41472.
a(n) = n^4*A013955(n).
Dirichlet g.f.: zeta(s-4)*zeta(s-11). - R. J. Mathar, Aug 03 2025

A282332 Coefficients in q-expansion of E_4^3*E_6^2, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

Original entry on oeis.org

1, -288, -325728, 11700864, 35176468896, 6601058210880, 438061091013504, 15173572442740992, 327251435243536800, 4913611331706352224, 55439979246339307200, 496425441863436557184, 3672747479405396310912, 23148319784349233726784

Offset: 0

Seiichi Manyama, Feb 12 2017

sign

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

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

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

A282328 Coefficients in q-expansion of E_4*E_6^3, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

Original entry on oeis.org

1, -1272, 351432, 89559456, -28689603384, -3415837464144, -155926897275744, -3967939206760128, -65540990858009400, -777517458842153496, -7105797244669716432, -52584588767807410464, -326903749149928526688, -1755591468945924647184

Offset: 0

Seiichi Manyama, Feb 12 2017

sign

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

Cf. A004009 (E_4), A013973 (E_6).

Cf. A013974 (E_4*E_6 = E_10), A282287 (E_4*E_6^2), this sequence (E_4*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]*E6[x]^3 + 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

sign

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 *)

A282597 Expansion of phi_{14, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

Original entry on oeis.org

0, 1, 16386, 4782972, 268468228, 6103515630, 78373779192, 678223072856, 4398583447560, 22876806803877, 100012207113180, 379749833583252, 1284076017413616, 3937376385699302, 11113363271818416, 29192944359852360, 72066391204823056, 168377826559400946

Offset: 0

Seiichi Manyama, Feb 19 2017

Multiplicative because A013961 is. - Andrew Howroyd, Jul 25 2018

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

Cf. A064987 (phi_{2, 1}), A281372 (phi_{4, 1}), A282050 (phi_{6, 1}), A282060 (phi_{8, 1}), A282254 (phi_{10, 1}), A282548 (phi_{12, 1}), this sequence (phi_{14, 1}).
Cf. A282012 (E_4^4), A282287 (E_4*E_6^2), A282596 (E_2*E_4^2*E_6).
Cf. A013672.

Table[n * DivisorSigma[13, n], {n, 0, 17}] (* Amiram Eldar, Sep 06 2023 *)

a(n) = if(n < 1, 0, n*sigma(n, 13)) \\ Andrew Howroyd, Jul 25 2018

a(n) = n*A013961(n) for n > 0.
a(n) = (3*A282012(n) + 4*A282287(n) - 7*A282596(n))/144.
Sum_{k=1..n} a(k) ~ zeta(14) * n^15 / 15. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^e * (p^(13*e+13)-1)/(p^13-1).
Dirichlet g.f.: zeta(s-1)*zeta(s-14). (End)

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

sign

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

sign

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

A029829 Eisenstein series E_16(q) (alternate convention E_8(q)), multiplied by 3617.

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A386787 a(n) = n^4*sigma_7(n).

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A282332 Coefficients in q-expansion of E_4^3*E_6^2, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

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A282328 Coefficients in q-expansion of E_4*E_6^3, 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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A282597 Expansion of phi_{14, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

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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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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.