A004009 -id:A004009 - 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

A055747 Expansion of Jacobi form of weight 12 and index 1 for the Niemeier lattice of type E_8^3 or D_16+E_8.

Original entry on oeis.org

1, 0, 0, 56, 606, 0, 0, 27456, 123156, 0, 0, 3745512, 9217112, 0, 0, 95209152, 188066718, 0, 0, 1144371624, 1960489800, 0, 0, 8505838656, 13289979912, 0, 0, 45755357024, 67080028224, 0, 0, 195411318912, 272570040468, 0, 0

Offset: 0

Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jul 11 2000

nonn

a(4*n-r^2) gives number of vectors x in the lattice of norm 2n and =r for any fixed vector in the lattice of norm 2.

M. Eichler and D. Zagier, The Theory of Jacobi Forms, Birkhauser, 1985.

Cf. A008411, A003783, A004009.

E_8*E_4, 1.

G.f.: b(z) * c(z) where b(z) is g.f. for A003783 and c(z) = 1 + 240*z^4 + 2160*z^8 + ... is A004009 expanded in powers of z^4. - Sean A. Irvine, Apr 05 2022

A289209 Coefficients in expansion of E_4^3/E_6^2.

Original entry on oeis.org

1, 1728, 1700352, 1332930816, 939690602496, 624182333927040, 399031077924476928, 248370528839869094400, 151578005556161702559744, 91116938989182168182098368, 54119528875319902426524072960, 31833210323194251819350736777984

Offset: 0

Seiichi Manyama, Jun 28 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..365

(E_4^3/E_6^2)^(k/288): A289365 (k=1), A299694 (k=2), A299696 (k=3), A299697 (k=4), A299698 (k=6), A299943 (k=8), A299949 (k=9), A289369 (k=12), A299950 (k=16), A299951 (k=18), A299953 (k=24), A299993 (k=32), A299994 (k=36), A300052 (k=48), A300053 (k=72), A300054 (k=96), A300055 (k=144), this sequence (k=288).
Cf. A004009 (E_4), A013973 (E_6), A289063, A289210, A289417, A300025.
E_{k+2}/E_k: A288261 (k=4, 8), A288840 (k=6).

nmax = 20; CoefficientList[Series[(1 + 240*Sum[DivisorSigma[3,k]*x^k, {k, 1, nmax}])^3 / (1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^2, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

G.f.: 1 + 1728 * q * Product_{k>=1} (1-q^k)^24 / E_6^2.
G.f.: (E_4*E_8)/(E_6*E_6) = (E_8*E_8)/(E_6*E_10). - Seiichi Manyama, Jun 29 2017
a(n) = 1728 * A289417(n - 1) for n > 0. - Seiichi Manyama, Jul 08 2017
a(n) ~ c * exp(2*Pi*n) * n, where c = 256 * Pi^6 / (3 * Gamma(1/4)^8) = 2.747700206704861755142526128354171788550012833617513654955480535522... - Vaclav Kotesovec, Jul 08 2017, updated Mar 04 2018
a(0) = 1, a(n) = (288/n)*Sum_{k=1..n} A300025(k)*a(n-k) for n > 0. - Seiichi Manyama, Feb 26 2018

A289210 Coefficients in expansion of E_6^2/E_4^3.

Original entry on oeis.org

1, -1728, 1285632, -616294656, 242544070656, -85253786824320, 27846073156184064, -8638345400999827968, 2579332695698905989120, -747814048389765750131136, 211795259563761765262894080, -58852853362216364363212075776

Offset: 0

Seiichi Manyama, Jun 28 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..420

(E_6^2/E_4^3)^(k/288): A289366 (k=1), A296609 (k=2), A296614 (k=3), A296652 (k=4), A297021 (k=6), A299422 (k=8), A299862 (k=9), A289368 (k=12), A299856 (k=16), A299857 (k=18), A299858 (k=24), A299863 (k=32), A299859 (k=36), A299860 (k=48), A299861 (k=72), A299414 (k=96), A299413 (k=144), this sequence (k=288).
Cf. A000521 (j), A004009 (E_4), A013973 (E_6), A066395, A289209, A300025.
E_{k+2}/E_k: A288261 (k=4, 8), A288840 (k=6).

nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^2 / (1 + 240*Sum[DivisorSigma[3,k]*x^k, {k, 1, nmax}])^3, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

a(n) = -1728 * A066395(n) for n > 0.
G.f.: 1 - 1728 * q * Product_{k>=1} (1-q^k)^24 / E_4^3 = 1 - 1728/j.
G.f.: (E_6*E_6)/(E_4*E_8) = (E_6*E_10)/(E_8*E_8). - Seiichi Manyama, Jun 29 2017
a(n) ~ (-1)^n * c * exp(Pi*sqrt(3)*n) * n^2, where c = 256 * Pi^12 / Gamma(1/3)^18 = 4.684993039417145659090436569582265840407909701042523126716193567422... - Vaclav Kotesovec, Jul 08 2017, updated Mar 04 2018
a(0) = 1, a(n) = -(288/n)*Sum_{k=1..n} A300025(k)*a(n-k) for n > 0. - Seiichi Manyama, Feb 26 2018

A288261 Coefficients in expansion of E_6/E_4.

Original entry on oeis.org

1, -744, 159768, -36866976, 8507424792, -1963211493744, 453039686271072, -104545516658693952, 24125403112135458840, -5567288717204029449672, 1284733088879405339418768, -296470902355240575283208928, 68414985730612787485819011168

Offset: 0

Seiichi Manyama, Jun 17 2017

sign

Also coefficients in expansion of E_10/E_8. - Seiichi Manyama, Jun 20 2017

			G.f.: 1 - 744*q + 159768*q^2 - 36866976*q^3 + 8507424792*q^4 - 1963211493744*q^5 + 453039686271072*q^6 + ...
From _Seiichi Manyama_, Jun 27 2017: (Start)
a(0) = j_0((-1+sqrt(3)*i)/2) = 1,_
a(1) = j_1((-1+sqrt(3)*i)/2) = -744 + 0^1 = -744,
a(2) = j_2((-1+sqrt(3)*i)/2) = 159768 - 1488*0^1 + 0^2 = 159768. (End)

Seiichi Manyama, Table of n, a(n) for n = 0..422

Cf. A004009 (E_4), A110163, A013973 (E_6).
E_{k+2}/E_k: A288877 (k=2), this sequence (k=4, 8), A288840 (k=6).
Cf. A000521 (j), A035230 (-q*j'), A066395 (1/j), A289141.

nmax = 20; CoefficientList[Series[(1 - 504*Sum[DivisorSigma[5, k]*x^k, {k, 1, nmax}])/(1 + 240*Sum[DivisorSigma[3, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 28 2017 *)
terms = 13; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[Ei[6]/Ei[4] + O[x]^terms, x] (* Jean-François Alcover, Mar 01 2018 *)
a[ n_] := With[{j = Series[1728 KleinInvariantJ[ Log[ Series[q, {q, 0, n + 1}]]/(2 Pi I)], {q, 0, n}]}, SeriesCoefficient[ -q D[j, q] / j, {q, 0, n}]]; (* Michael Somos, Aug 15 2018 *)

From Seiichi Manyama, Jun 27 2017: (Start)
Let j_0 = 1 and j_1 = j - 744. Define j_m by j_m = j1 | T_0(m), where T_0(m) = mT_{m, 0} is the normalized m-th weight zero Hecke operator. a(n) = j_n((-1+sqrt(3)*i)/2).
G.f.: Sum_{n >= 0} j_n((-1+sqrt(3)*i)/2)*q^n. (End)
a(n) ~ (-1)^n * 3 * exp(Pi*sqrt(3)*n). - Vaclav Kotesovec, Jun 28 2017
G.f.: -q*j'/j where j is the elliptic modular invariant (A000521). - Seiichi Manyama, Jul 12 2017

A299694 Coefficients in expansion of (E_4^3/E_6^2)^(1/144).

Original entry on oeis.org

1, 12, 1512, 813744, 281434656, 129501949608, 56296822560480, 26218237904433888, 12242575532254540032, 5850239653863742634172, 2820869122426120317439152, 1375631026432164061822527120, 675950202173640832786529615232

Offset: 0

Seiichi Manyama, Feb 16 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..368

(E_4^3/E_6^2)^(k/288): A289365 (k=1), this sequence (k=2), A299696 (k=3), A299697 (k=4), A299698 (k=6), A299943 (k=8), A299949 (k=9), A289369 (k=12), A299950 (k=16), A299951 (k=18), A299953 (k=24), A299993 (k=32), A299994 (k=36), A300052 (k=48), A300053 (k=72), A300054 (k=96), A300055 (k=144), A289209 (k=288).

Cf. A004009 (E_4), A013973 (E_6), A296609.

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]^3/E6[x]^2)^(1/144) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Convolution inverse of A296609.
a(n) ~ 2^(1/18) * Pi^(1/24) * exp(2*Pi*n) / (3^(1/144) * Gamma(1/72) * Gamma(1/4)^(1/18) * n^(71/72)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A296609(n) ~ -sin(Pi/72) * exp(4*Pi*n) / (72*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A299696 Coefficients in expansion of (E_4^3/E_6^2)^(1/96).

Original entry on oeis.org

1, 18, 2322, 1234116, 430292646, 197681749128, 86165040337452, 40145493017336976, 18768723217958523222, 8975036477140737601806, 4331009172188712335053032, 2113419430011730408087143924, 1039122180212218474089489166980

Offset: 0

Seiichi Manyama, Feb 16 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..368

(E_4^3/E_6^2)^(k/288): A289365 (k=1), A299694 (k=2), this sequence (k=3), A299697 (k=4), A299698 (k=6), A299943 (k=8), A299949 (k=9), A289369 (k=12), A299950 (k=16), A299951 (k=18), A299953 (k=24), A299993 (k=32), A299994 (k=36), A300052 (k=48), A300053 (k=72), A300054 (k=96), A300055 (k=144), A289209 (k=288).

Cf. A004009 (E_4), A013973 (E_6), A296614.

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]^3/E6[x]^2)^(1/96) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Convolution inverse of A296614.
a(n) ~ 2^(1/12) * Pi^(1/16) * exp(2*Pi*n) / (3^(1/96) * Gamma(1/48) * Gamma(1/4)^(1/12) * n^(47/48)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A296614(n) ~ -sin(Pi/48) * exp(4*Pi*n) / (48*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A299697 Coefficients in expansion of (E_4^3/E_6^2)^(1/72).

Original entry on oeis.org

1, 24, 3168, 1663776, 584685312, 268219092816, 117214929608832, 54637244971358016, 25574598700199847936, 12238100148358426410360, 5910293921259795914011968, 2885917219371433467109558368, 1419817980186833008095972357120

Offset: 0

Seiichi Manyama, Feb 16 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..367

(E_4^3/E_6^2)^(k/288): A289365 (k=1), A299694 (k=2), A299696 (k=3), this sequence (k=4), A299698 (k=6), A299943 (k=8), A299949 (k=9), A289369 (k=12), A299950 (k=16), A299951 (k=18), A299953 (k=24), A299993 (k=32), A299994 (k=36), A300052 (k=48), A300053 (k=72), A300054 (k=96), A300055 (k=144), A289209 (k=288).

Cf. A004009 (E_4), A013973 (E_6), A296652.

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]^3/E6[x]^2)^(1/72) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Convolution inverse of A296652.
a(n) ~ 2^(1/9) * Pi^(1/12) * exp(2*Pi*n) / (3^(1/72) * Gamma(1/36) * Gamma(1/4)^(1/9) * n^(35/36)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A296652(n) ~ -sin(Pi/36) * exp(4*Pi*n) / (36*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A299698 Coefficients in expansion of (E_4^3/E_6^2)^(1/48).

Original entry on oeis.org

1, 36, 4968, 2551824, 910405152, 416585268216, 182967944992992, 85373023607994528, 40055910812083687680, 19194979975339075406388, 9284600439037161721276848, 4539375955473797523355108272, 2236041702620444573315950439808

Offset: 0

Seiichi Manyama, Feb 16 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..367

(E_4^3/E_6^2)^(k/288): A289365 (k=1), A299694 (k=2), A299696 (k=3), A299697 (k=4), this sequence (k=6), A299943 (k=8), A299949 (k=9), A289369 (k=12), A299950 (k=16), A299951 (k=18), A299953 (k=24), A299993 (k=32), A299994 (k=36), A300052 (k=48), A300053 (k=72), A300054 (k=96), A300055 (k=144), A289209 (k=288).

Cf. A004009 (E_4), A013973 (E_6), A297021.

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]^3/E6[x]^2)^(1/48) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

Convolution inverse of A297021.
a(n) ~ 2^(1/6) * Pi^(1/8) * exp(2*Pi*n) / (3^(1/48) * Gamma(1/24) * Gamma(1/4)^(1/6) * n^(23/24)). - Vaclav Kotesovec, Mar 04 2018
a(n) * A297021(n) ~ -sin(Pi/24) * exp(4*Pi*n) / (24*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

A299943 Coefficients in expansion of (E_4^3/E_6^2)^(1/36).

Original entry on oeis.org

1, 48, 6912, 3479616, 1259268096, 575044765344, 253777092387840, 118545813515338368, 55748828845833043968, 26753648919849657887472, 12960874757914028815661568, 6344939709971525751086888640, 3129285552537639403735326646272

Offset: 0

Seiichi Manyama, Feb 22 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..367

(E_4^3/E_6^2)^(k/288): A289365 (k=1), A299694 (k=2), A299696 (k=3), A299697 (k=4), A299698 (k=6), this sequence (k=8), A299949 (k=9), A289369 (k=12), A299950 (k=16), A299951 (k=18), A299953 (k=24), A299993 (k=32), A299994 (k=36), A300052 (k=48), A300053 (k=72), A300054 (k=96), A300055 (k=144), A289209 (k=288).

Cf. A004009 (E_4), A013973 (E_6), A299422.

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]^3/E6[x]^2)^(1/36) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 28 2018 *)

Convolution inverse of A299422.
a(n) ~ c * exp(2*Pi*n) / n^(17/18), where c = 2^(2/9) * Pi^(1/6) / (3^(1/36) * Gamma(1/4)^(2/9) * Gamma(1/18)) = 0.0588537525900341685779220592527938... - Vaclav Kotesovec, Mar 04 2018
a(n) * A299422(n) ~ -sin(Pi/18) * exp(4*Pi*n) / (18*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018

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).

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A055747 Expansion of Jacobi form of weight 12 and index 1 for the Niemeier lattice of type E_8^3 or D_16+E_8.

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A289209 Coefficients in expansion of E_4^3/E_6^2.

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A289210 Coefficients in expansion of E_6^2/E_4^3.

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A288261 Coefficients in expansion of E_6/E_4.

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A299694 Coefficients in expansion of (E_4^3/E_6^2)^(1/144).

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A299696 Coefficients in expansion of (E_4^3/E_6^2)^(1/96).

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A299697 Coefficients in expansion of (E_4^3/E_6^2)^(1/72).

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A299698 Coefficients in expansion of (E_4^3/E_6^2)^(1/48).