A280869 -id:A280869 - cp's OEIS frontend

A008411 Theta series of direct sum of 3 copies of E_8 lattice (the Niemeier lattice of type E_8^3).

1, 720, 179280, 16954560, 396974160, 4632858720, 34413301440, 187477879680, 814940600400, 2975469665040, 9486467837280, 27053330840640, 70485969919680, 169930679355360, 384163875688320, 820167497170560, 1668890801059920, 3249626139960480, 6096884624994960

Offset: 0

N. J. A. Sloane

Also the theta series for the Niemeier lattice of type E_8 D_16. - Ben Mares, Jul 17 2022

			G.f. = 1 + 720*q + 179280*q^2 + 16954560*q^3 + 396974160*q^4 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 123, 407.

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

Cf. A000594, A004009, A029828, A280869.

A := Basis( ModularForms( Gamma1(1), 12), 19); A[1] + 720*A[2]; /* Michael Somos, Jan 28 2017 */

a[ n_] := SeriesCoefficient[ With[ {t2 = EllipticTheta[ 2, 0, q]^4, t3 = EllipticTheta[ 3, 0, q]^4}, (t2^2 + 14 t2 t3 + t3^2)^3 ], {q, 0, n}]; (* Michael Somos, Jan 28 2017 *)
terms = 19; QP = QPochhammer; s = (QP[x]^24 + 256*x*QP[x^2]^24)^3 / (QP[x]*QP[x^2])^24 + O[x]^terms; CoefficientList[s, x] (* Jean-François Alcover, Jul 07 2017, adapted from PARI *)
terms = 19; E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}]; E4[x]^3 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^24 + 256 * x * eta(x^2 + A)^24)^3 / (eta(x + A) * eta(x^2 + A))^24, n))}; /* Michael Somos, Jan 28 2017 */

This series is the q-expansion of E_4(z)^3. Cf. A004009. - Daniel D. Briggs, Nov 25 2011

691*a(n) - A029828(n) = 432000*A000594(n). - Seiichi Manyama, Jan 28 2017

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

Original entry on oeis.org

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.

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 *)

A280025 Expansion of phi_{7, 4}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

Original entry on oeis.org

0, 1, 144, 2268, 18688, 78750, 326592, 825944, 2396160, 4966677, 11340000, 19501812, 42384384, 62777078, 118935936, 178605000, 306774016, 410422194, 715201488, 894002060, 1471680000, 1873240992, 2808260928, 3405105288, 5434490880, 6152734375, 9039899232

Offset: 0

Seiichi Manyama, Feb 22 2017

Multiplicative because A001158 is. - Andrew Howroyd, Jul 23 2018

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

Cf. A280022 (phi_{5, 4}), this sequence (phi_{7, 4}).
Cf. A280024 (E_2^4*E_4), A282780 (E_2^3*E_6), A282752 (E_2^2*E_4^2), A282102 (E_2*E_4*E_6), A008411 (E_4^3), A280869 (E_6^2).
Cf. A001158 (sigma_3(n)), A281372 (n*sigma_3(n)), A282099 (n^2*sigma_3(n)), A282213 (n^3*sigma_3(n)), this sequence (n^4*sigma_3(n)).
Cf. A152649.

Table[n^4 * DivisorSigma[3, n], {n, 0, 30}] (* Amiram Eldar, Oct 31 2023 *)
nmax = 30; CoefficientList[Series[Sum[k^4*x^k*(1 + 120*x^k + 1191*x^(2*k) + 2416*x^(3*k) + 1191*x^(4*k) + 120*x^(5*k) + x^(6*k))/(1 - x^k)^8, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 02 2025 *)

a(n) = if(n < 1, 0, n^4 * sigma(n, 3)); \\ Andrew Howroyd, Jul 23 2018

a(n) = n^4*A001158(n) for n > 0.
a(n) = (7*(A280024(n) - 4*A282780(n) + 6*A282752(n) - 4*A282102(n)) + 3*A008411(n) + 4*A280869(n))/41472.
Sum_{k=1..n} a(k) ~ c * n^8, where c = Pi^4/720 = 0.1352904... (= A152649 / 10). - Amiram Eldar, Dec 08 2022
From Amiram Eldar, Oct 31 2023: (Start)
Multiplicative with a(p^e) = p^(4*e) * (p^(3*e+3)-1)/(p^3-1).
Dirichlet g.f.: zeta(s-4)*zeta(s-7). (End)
G.f.: Sum_{k>=1} k^4*x^k*(1 + 120*x^k + 1191*x^(2*k) + 2416*x^(3*k) + 1191*x^(4*k) + 120*x^(5*k) + x^(6*k))/(1 - x^k)^8. - Vaclav Kotesovec, Aug 02 2025

A282253 Coefficients in q-expansion of E_6^3, where E_6 is the Eisenstein series shown in A013973.

Original entry on oeis.org

1, -1512, 712152, -78097824, -11474230824, -498089967984, -11088580243104, -152351956669248, -1474676091461160, -10921529499813576, -65490182325115632, -331010378444247264, -1452953351890984608, -5665062963045803184, -19968586384352171328

Offset: 0

Seiichi Manyama, Feb 10 2017

sign

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

Cf. A282018 (E_2^3), A008411 (E_4^3), this sequence (E_6^3).

Cf. A013973 (E_6), A280869 (E_6^2), this sequence (E_6^3).

terms = 15;
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E6[x]^3 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

A282254 Coefficients in q-expansion of (3E_4^3 + 2E_6^2 - 5E_2E_4*E_6)/1584, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

0, 1, 1026, 59052, 1050628, 9765630, 60587352, 282475256, 1075843080, 3486961557, 10019536380, 25937424612, 62041684656, 137858491862, 289819612656, 576679982760, 1101663313936, 2015993900466, 3577622557482, 6131066257820, 10260044315640

Offset: 0

Seiichi Manyama, Feb 10 2017

Multiplicative because A013957 is. - Andrew Howroyd, Jul 25 2018

D. H. Lehmer shows that a(n) == tau(n) (mod 7) for n > 0, where tau is Ramanujan's tau function (A000594). Furthermore, if n == 3, 5, 6 (mod 7) then a(n) == tau(n) (mod 49). See the Wikipedia link below. - Jianing Song, Aug 12 2020

			a(6) = 1^10*6^1 + 2^10*3^1 + 3^10*2^1 + 6^10*1^1 = 60587352.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Wikipedia, Congruences for the tau function.

Cf. A064987 (phi_{2, 1}), A281372 (phi_{4, 1}), A282050 (phi_{6, 1}), A282060 (phi_{8, 1}), this sequence (phi_{10, 1}).
Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A008411 (E_4^3), A280869 (E_6^2), A282102 (E_2*E_4*E_6).
Cf. A013957, A013668, A196292.

Table[If[n>0, n * DivisorSigma[9, n], 0], {n, 0, 20}] (* Indranil Ghosh, Mar 12 2017 *)

for(n=0, 20, print1(if(n==0, 0, n * sigma(n, 9)),", ")) \\ Indranil Ghosh, Mar 12 2017

G.f.: phi_{10, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
a(n) = (3*A008411(n) + 2*A280869(n) - 5*A282102(n))/1584.
If p is a prime, a(p) = p^10 + p = A196292(p).
a(n) = n*A013957(n) for n > 0, where A013957(n) is sigma_9(n), the sum of the 9th powers of the divisors of n. - Seiichi Manyama, Feb 18 2017
Multiplicative with a(p^e) = p^e*(p^(9*(e+1))-1)/(p^9-1). - Jianing Song, Aug 12 2020
From Amiram Eldar, Oct 30 2023: (Start)
Dirichlet g.f.: zeta(s-1)*zeta(s-10).
Sum_{k=1..n} a(k) ~ zeta(10) * n^11 / 11. (End)

A282753 Expansion of phi_{9, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

Original entry on oeis.org

0, 1, 516, 19692, 264208, 1953150, 10161072, 40353656, 135274560, 387597717, 1007825400, 2357947812, 5202783936, 10604499542, 20822486496, 38461429800, 69260574976, 118587876786, 200000421972, 322687698140, 516037855200, 794644193952, 1216701070992

Offset: 0

Seiichi Manyama, Feb 21 2017

Multiplicative because A013955 is. - Andrew Howroyd, Jul 25 2018

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

Cf. A282097 (phi_{3, 2}), A282099 (phi_{5, 2}), A282751 (phi_{7, 2}), this sequence (phi_{9, 2}).
Cf. A282752 (E_2^2*E_4^2), A282102 (E_2*E_4*E_6), A008411 (E_4^3), A280869 (E_6^2).
Cf. A013955 (sigma_7(n)), A282060 (n*sigma_7(n)), this sequence (n^2*sigma_7(n)).
Cf. A013666.

Table[If[n>0, n^2 * DivisorSigma[7, n], 0], {n, 0, 22}] (* Indranil Ghosh, Mar 12 2017 *)
nmax = 40; CoefficientList[Series[Sum[k^9*x^k*(1 + x^k)/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 02 2025 *)

for(n=0, 22, print1(if(n==0, 0, n^2 * sigma(n, 7)),", ")) \\ Indranil Ghosh, Mar 12 2017

a(n) = n^2*A013955(n) for n > 0.
a(n) = (9*A282752(n) - 18*A282102(n) + 5*A008411(n) + 4*A280869(n))/8640.
Sum_{k=1..n} a(k) ~ zeta(8) * n^10 / 10. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^(2*e) * (p^(7*e+7)-1)/(p^7-1).
Dirichlet g.f.: zeta(s-2)*zeta(s-9). (End)
G.f.: Sum_{k>=1} k^9*x^k*(1 + x^k)/(1 - x^k)^3. - Vaclav Kotesovec, Aug 02 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 *)

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

A377975 Expansion of the 6048th root of the series 2*E_6(x) - E_6(x)^2, where E_6 is the Eisenstein series of weight 6.

Original entry on oeis.org

1, 0, -42, -2772, -5399688, -704781084, -943173698460, -180121119486672, -188146584694894350, -46293152603021155692, -40574254265781269371884, -11963000065787771567311500, -9221266403646163252100062068, -3107813621461888912485774582588, -2176998806586925223600540321844120

Offset: 0

Peter Bala, Nov 14 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_6(x) lies in P(12) (Heninger et al.).
We claim that the series 2*E_6(x) - E_6(x)^2 belongs to P(6048).
Proof.
E_6(x) = 1 - 504*Sum_{n >= 1} sigma_5(n)*x^n. Hence,
2*E_6(x) - E_6(x)^2 = 1 - (504^2)*( Sum_{n >= 1} sigma_5(n)*x^n )^2 is in R.
Hence, 2*E_6(x) - E_6(x)^2 == 1 (mod 504^2) == 1 (mod (2^6)*(3^4)*(7^2)).
It follows from Heninger et al., Theorem 1, Corollary 2, that the series 2*E_6(x) - E_6(x)^2 belongs to P((2^5)*(3^3)*7) = P(6048). End Proof.

Cf. A013973 (E_6), A109817 ( (E_6)^1/12 ), A280869 (E_6)^2, A341871 - A341875, A377973, A377974, 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(6) - E(6)^2)^(1/6048), q = 0, n),n = 0..20);

terms = 20; E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}]; CoefficientList[Series[(2*E6[x] - E6[x]^2)^(1/6048), {x, 0, terms}], x] (* Vaclav Kotesovec, Aug 03 2025 *)

a(n) ~ c / (r^n * n^(6049/6048)), where r = 0.0018674427317079888144302129348270303934228050024753171993815386383179351229... is the root of the equation Sum_{k>=1} sigma_5(k) * r^k = 1/504 and c = -0.0001653486643613776568861731992670297686378824546... - Vaclav Kotesovec, Aug 03 2025

cp's OEIS Frontend

A008411 Theta series of direct sum of 3 copies of E_8 lattice (the Niemeier lattice of type E_8^3).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A280025 Expansion of phi_{7, 4}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A282253 Coefficients in q-expansion of E_6^3, where E_6 is the Eisenstein series shown in A013973.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A282254 Coefficients in q-expansion of (3*E_4^3 + 2*E_6^2 - 5*E_2*E_4*E_6)/1584, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A282753 Expansion of phi_{9, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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.

Views

Author

Keywords

A282254 Coefficients in q-expansion of (3E_4^3 + 2E_6^2 - 5E_2E_4*E_6)/1584, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.