A008411 -id:A008411 - cp's OEIS frontend

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.

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

sign

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 *)

A288846 Expansion of (q*j)^3, where j is a modular function A000521.

Original entry on oeis.org

1, 2232, 2251260, 1355202240, 541778118390, 151522053809760, 30456116651640888, 4460775211418664960, 479919718908048515625, 38292247221915373896560, 2309356967925215526546564, 108570959012192293978767360, 4111854826236389868361040550

Offset: 0

Seiichi Manyama, Jun 18 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)

Cf. A000521 (j(q)), A004009 (E_4), A008411 (E_4^3).

(q*j(q))^(k/24): A289397 (k=-1), A106205 (k=1), A289297 (k=2), A289298 (k=3), A289299 (k=4), A289300 (k=5), A289301 (k=6), A289302 (k=7), A007245 (k=8), A289303 (k=9), A289304 (k=10), A289305 (k=11), A161361 (k=12), A028512 (k=16), A028513 (k=32), A028514 (k=40), A028515 (k=48), this sequence (k=72).

CoefficientList[Series[(QPochhammer[x, x^2]^8 + 256*x/QPochhammer[x, x^2]^16)^9, {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 29 2017 *)
(q*1728*KleinInvariantJ[-Log[q]*I/(2*Pi)])^3 + O[q]^13 // CoefficientList[#, q]& (* Jean-François Alcover, Nov 02 2017 *)

G.f.: ((1 + 240 Sum_{k>0} k^3 q^k/(1-q^k))^3/(Product_{k>0} (1-q^k)^24))^3.

a(n) ~ 3^(1/4) * exp(4*Pi*sqrt(3*n)) / (sqrt(2)*n^(3/4)). - Vaclav Kotesovec, Jun 29 2017

A282330 Coefficients in q-expansion of E_4^6, where E_4 is the Eisenstein series A004009.

Original entry on oeis.org

1, 1440, 876960, 292072320, 57349833120, 6660135541440, 436536302762880, 15172132360815360, 327295477379498400, 4913576699608450080, 55439481453769056960, 496426192564963006080, 3672749219557161663360, 23148323907214334109120

Offset: 0

Seiichi Manyama, Feb 12 2017

nonn

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

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

terms = 14;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E4[x]^6 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

G.f.: (1 + 240 Sum_{i>=1} i^3 q^i/(1-q^i))^6.

A282781 Expansion of phi_{8, 3}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

Original entry on oeis.org

0, 1, 264, 6588, 67648, 390750, 1739232, 5765144, 17318400, 43224597, 103158000, 214360212, 445665024, 815732918, 1521998016, 2574261000, 4433514496, 6975762354, 11411293608, 16983569900, 26433456000, 37980768672, 56591095968, 78310997448

Offset: 0

Seiichi Manyama, Feb 21 2017

Multiplicative because A001160 is. - Andrew Howroyd, Jul 25 2018

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

Cf. A282211 (phi_{4, 3}), A282213 (phi_{6, 3}), this sequence (phi_{8, 3}).
Cf. A282752 (E_2^2*E_4^2), A282780 (E_2^3*E_6), A282102 (E_2*E_4*E_6), A008411 (E_4^3), A280869 (E_6^2).
Cf. A001160 (sigma_5(n)), A282050 (n*sigma_5(n)), A282751 (n^2*sigma_5(n)), this sequence (n^3*sigma_5(n)).
Cf. A013664.

a[0]=0;a[n_]:=(n^3)*DivisorSigma[5,n];Table[a[n],{n,0,23}] (* Indranil Ghosh, Feb 21 2017 *)
nmax = 30; CoefficientList[Series[Sum[k^8*x^k*(x^(2*k) + 4*x^k + 1)/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 02 2025 *)

a(n) = if (n==0, 0, n^3*sigma(n, 5)); \\ Michel Marcus, Feb 21 2017

a(n) = n^3*A001160(n) for n > 0.
a(n) = (6*A282752(n) - 2*A282780(n) - 6*A282102(n) + A008411(n) + A280869(n))/5184.
Sum_{k=1..n} a(k) ~ zeta(6) * n^9 / 9. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 31 2023: (Start)
Multiplicative with a(p^e) = p^(3*e) * (p^(5*e+5)-1)/(p^5-1).
Dirichlet g.f.: zeta(s-3)*zeta(s-8). (End)
G.f.: Sum_{k>=1} k^8*x^k*(x^(2*k) + 4*x^k + 1)/(1 - x^k)^4. - Vaclav Kotesovec, Aug 02 2025

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

Original entry on oeis.org

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.).

A282402 Coefficients in q-expansion of E_4^7, where E_4 is the Eisenstein series A004009.

Original entry on oeis.org

1, 1680, 1224720, 505659840, 129351117840, 21060890131680, 2160822606183360, 134717272385473920, 4957295423282269200, 119288258695393463760, 2051465861242156554720, 26894077218337493424960, 281803532524538902825920

Offset: 0

Seiichi Manyama, Feb 14 2017

sign

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), this sequence (E_4^7).

terms = 13;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E4[x]^7 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 27 2018 *)

A290009 Coefficients in expansion of 691E_4E_8*E_12.

Original entry on oeis.org

691, 563040, 305307360, 131729109120, 34085393629920, 4587384326495040, 302027782271806080, 10484303481804821760, 226150164335242994400, 3395290157453914541280, 38308806132696980919360, 343030311387007824977280, 2537869275676057371269760

Offset: 0

Seiichi Manyama, Jul 17 2017

nonn

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

Cf. A004009 (E_4), A008410 (E_8), A008411 (E_4^3), A029828 (691*E_12).

Cf. A290010.

terms = 13;
E4[x_] = 1 + 240*Sum[k^3*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*E4[x]^3*E12[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

G.f.: 691*E_4^3*E_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.

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

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 *)

A299955 Coefficients in expansion of E_4^(3/2).

Original entry on oeis.org

1, 360, 24840, -465120, 57417480, -6800282640, 930889890720, -139401582644160, 22250341370421000, -3723955494287559480, 646515765251485521840, -115559140273640812421280, 21150946022800731753255840, -3948247836773858791840263120

Offset: 0

Seiichi Manyama, Feb 22 2018

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Seiichi Manyama, Table of n, a(n) for n = 0..426

E_4^(k/8): A108091 (k=1), A289307 (k=2), A289308 (k=3), A289292 (k=4), A289309 (k=5), A289318 (k=6), A289319 (k=7), A004009 (k=8), this sequence (k=12), A008410 (k=16), A008411 (k=24), A282012 (k=32), A282015 (k=40).

Convolution cube of A289292.

a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) / n^(5/2), where c = 81*Gamma(1/3)^27 / (32768*sqrt(2)*Pi^(37/2)) = 0.39832876770813443250501819621900549862424768734... - Vaclav Kotesovec, Mar 05 2018

cp's OEIS Frontend

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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A288846 Expansion of (q*j)^3, where j is a modular function A000521.

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A282330 Coefficients in q-expansion of E_4^6, where E_4 is the Eisenstein series A004009.

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A282781 Expansion of phi_{8, 3}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

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

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A282402 Coefficients in q-expansion of E_4^7, where E_4 is the Eisenstein series A004009.

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A290009 Coefficients in expansion of 691*E_4*E_8*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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A282474 Coefficients in q-expansion of E_4^8, where E_4 is the Eisenstein series A004009.

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A290009 Coefficients in expansion of 691E_4E_8*E_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.