cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 15 results. Next

A008688 Theta series of Niemeier lattice of type D_24.

Original entry on oeis.org

1, 1104, 170064, 17051328, 396408912, 4634713440, 34410979008, 187471449984, 814973040720, 2975426026128, 9486423324000, 27053536131648, 70485827477184, 169930457503968, 384164030001024
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

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

Links

Crossrefs

Cf. A008689 - A008704.

Programs

  • Mathematica
    (* Coefficients 11/9 and -2/9 were computed from the data. *)
terms = 15; E4[q_] := 1 + 240 Sum[DivisorSigma[3, n]*q^(2 n), {n, 1, terms}]; E6[q_] := 1 - 504 Sum[DivisorSigma[5, n]*q^(2 n), {n, 1, terms}]; s = 11/9 E4[q]^3 - 2/9 E6[q]^2 + O[q]^(3 terms); Partition[ CoefficientList[s, q], 2][[All, 1]][[1 ;; terms]] (* Jean-François Alcover, Jul 06 2017 *)

A008703 Theta series of Niemeier lattice of type A_2^12.

Original entry on oeis.org

1, 72, 194832, 16791264, 397928016, 4629728880, 34417220544, 187488729792, 814885857360, 2975543305704, 9486542953440, 27052984412064, 70486210291392, 169931053729584, 384163615285632
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

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

Links

Crossrefs

Cf. A004009, A013973.
Cf. A008688 - A008702, A008704.

Programs

  • Mathematica
    terms = 15; E4[q_] := 1 + 240 Sum[DivisorSigma[3, n]*q^(2 n), {n, 1, terms}]; E6[q_] := 1 - 504 Sum[DivisorSigma[5, n]*q^(2 n), {n, 1, terms}]; s = 5/8 E4[q]^3 + 3/8 E6[q]^2 + O[q]^(3 terms); Partition[ CoefficientList[s, q], 2][[All, 1]][[1 ;; terms]] (* Jean-François Alcover, Jul 06 2017 *)

Formula

This series is the q-expansion of (5*E_4(z)^3 + 3*E_6(z)^2)/8. - Daniel D. Briggs, Nov 25 2011

A008695 Theta series of Niemeier lattice of type A_11 D_7 E_6.

Original entry on oeis.org

1, 288, 189648, 16845696, 397610064, 4630772160, 34415914176, 187485113088, 814904105040, 2975518758816, 9486517914720, 27053099888256, 70486130167488, 169930928938176, 384163702086528
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Also the theta series of the Niemeier lattice of type E_6^4. - clarified by Ben Mares, Sep 13 2022

References

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

Links

Crossrefs

Cf. A004009, A013973.
Cf. A008688 - A008694, A008696 - A008704.

Programs

  • Mathematica
    terms = 15; th = EllipticTheta; E4 = 1 + 240*Sum[k^3*(q^k/(1 - q^k)), {k, 1, terms}] + O[q]^terms; E6 = th[2, 0, q]^12 + th[3, 0, q]^12 - 33*th[2, 0, q]^4*th[3, 0, q]^4*(th[2, 0, q]^4 + th[3, 0, q]^4); CoefficientList[ (3/4)*E4^3 + (1/4)*E6^2 + O[q]^terms, q] (* Jean-François Alcover, Jul 05 2017 *)

Formula

This series is the q-expansion of (3*E_4(z)^3 + E_6(z)^2)/4. - Daniel D. Briggs, Nov 25 2011

A008696 Theta series of Niemeier lattice of type D_6^4.

Original entry on oeis.org

1, 240, 190800, 16833600, 397680720, 4630540320, 34416204480, 187485916800, 814900050000, 2975524213680, 9486523478880, 27053074226880, 70486147972800, 169930956669600, 384163682797440, 820166912933760
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Also the theta series for the Niemeier lattice of type A_9^2 D_6. - clarified by Ben Mares, Sep 13 2022

References

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

Links

Crossrefs

Cf. A004009, A013973.
Cf. A008688 - A008695, A008697 - A008704.

Programs

  • Mathematica
    terms = 15; th = EllipticTheta; E4 = 1 + 240*Sum[k^3*(q^k/(1 - q^k)), {k, 1, terms}] + O[q]^terms; E6 = th[2, 0, q]^12 + th[3, 0, q]^12 - 33*th[2, 0, q]^4*th[3, 0, q]^4*(th[2, 0, q]^4 + th[3, 0, q]^4); CoefficientList[ (13/18)*E4^3 + (5/18)*E6^2 + O[q]^terms, q] (* Jean-François Alcover, Jul 05 2017 *)

Formula

This series is the q-expansion of (13*E_4(z)^3 + 5*E_6(z)^2)/18. - Daniel D. Briggs, Nov 25 2011

A008700 Theta series of Niemeier lattice of type D_4^6.

Original entry on oeis.org

1, 144, 193104, 16809408, 397822032, 4630076640, 34416785088, 187487524224, 814891939920, 2975535123408, 9486534607200, 27053022904128, 70486183583424, 169931012132448, 384163644219264, 820166796086400
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Also the theta series of the Niemeier lattice of type A_5^4 D_4. - clarified by Ben Mares, Jul 17 2022

References

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

Links

Crossrefs

Cf. A004009, A013973.
Cf. A008688 - A008699, A008701, A008702, A008703, A008704.

Programs

  • Mathematica
    terms = 15; th = EllipticTheta; E4 = 1 + 240*Sum[k^3*(q^k/(1 - q^k)), {k, 1, terms}] + O[q]^terms; E6 = th[2, 0, q]^12 + th[3, 0, q]^12 - 33*th[2, 0, q]^4*th[3, 0, q]^4*(th[2, 0, q]^4 + th[3, 0, q]^4); CoefficientList[ (2/3)*E4^3 + (1/3)*E6^2 + O[q]^terms, q] (* Jean-François Alcover, Jul 05 2017 *)

Formula

This series is the q-expansion of (2*E_4(z)^3 + E_6(z)^2)/3. - Daniel D. Briggs, Nov 25 2011

A008702 Theta series of Niemeier lattice of type A_3^8.

Original entry on oeis.org

1, 96, 194256, 16797312, 397892688, 4629844800, 34417075392, 187488327936, 814887884880, 2975540578272, 9486540171360, 27052997242752, 70486201388736, 169931039863872, 384163624930176
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

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

Links

Crossrefs

Cf. A004009, A013973.
Cf. A008688 - A008701, A008703, A008704.

Programs

  • Mathematica
    terms = 15; E4[q_] := 1 + 240 Sum[DivisorSigma[3, n]*q^(2 n), {n, 1, terms}]; E6[q_] := 1 - 504 Sum[DivisorSigma[5, n]*q^(2 n), {n, 1, terms}]; s = 23/36 E4[q]^3 + 13/36 E6[q]^2 + O[q]^(3 terms); Partition[ CoefficientList[s, q], 2][[All, 1]][[1 ;; terms]] (* Jean-François Alcover, Jul 06 2017 *)

Formula

This series is the q-expansion of (23*E_4(z)^3 + 13*E_6(z)^2)/36. - Daniel D. Briggs, Nov 26 2011

A008693 Theta series of Niemeier lattice of type D_8^3.

Original entry on oeis.org

1, 336, 188496, 16857792, 397539408, 4631004000, 34415623872, 187484309376, 814908160080, 2975513303952, 9486512350560, 27053125549632, 70486112362176, 169930901206752, 384163721375616
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

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

Links

Crossrefs

Cf. A004009, A013973.
Cf. A008688 - A008692, A008694 - A008704.

Programs

  • Mathematica
    terms = 15; th = EllipticTheta; E4 = 1 + 240*Sum[k^3*(q^k/(1 - q^k)), {k, 1, terms}] + O[q]^terms; E6 = th[2, 0, q]^12 + th[3, 0, q]^12 - 33*th[2, 0, q]^4*th[3, 0, q]^4*(th[2, 0, q]^4 + th[3, 0, q]^4); CoefficientList[(7/9)*E4^3 + (2/9)*E6^2 + O[q]^terms, q] (* Jean-François Alcover, Jul 05 2017 *)

Formula

This series is the q-expansion of (7*E_4(z)^3 + 2*E_6(z)^2)/9. - Daniel D. Briggs, Nov 25 2011

A008694 Theta series of Niemeier lattice of type A_12^2.

Original entry on oeis.org

1, 312, 189072, 16851744, 397574736, 4630888080, 34415769024, 187484711232, 814906132560, 2975516031384, 9486515132640, 27053112718944, 70486121264832, 169930915072464, 384163711731072
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

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

Links

Crossrefs

Cf. A004009, A013973.
Cf. A008688 - A008693, A008695 - A008704.

Programs

  • Mathematica
    terms = 15; E4[q_] := 1 + 240 Sum[DivisorSigma[3, n]*q^(2 n), {n, 1, terms}]; E6[q_] := 1 - 504 Sum[DivisorSigma[5, n]*q^(2 n), {n, 1, terms}]; s = 55/72 E4[q]^3 + 17/72 E6[q]^2 + O[q]^(3 terms); Partition[ CoefficientList[s, q], 2][[All, 1]][[1 ;; terms]] (* Jean-François Alcover, Jul 06 2017 *)

Formula

This series is the q-expansion of (55*E_4(z)^3 + 17*E_6(z)^2)/72. - Daniel D. Briggs, Nov 25 2011

A008697 Theta series of Niemeier lattice of type A_8^3.

Original entry on oeis.org

1, 216, 191376, 16827552, 397716048, 4630424400, 34416349632, 187486318656, 814898022480, 2975526941112, 9486526260960, 27053061396192, 70486156875456, 169930970535312, 384163673152896
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

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

Links

Crossrefs

Cf. A004009, A013973.
Cf. A008688 - A008696, A008698 - A008704.

Programs

  • Mathematica
    terms = 15; E4[q_] := 1 + 240 Sum[DivisorSigma[3, n]*q^(2 n), {n, 1, terms}]; E6[q_] := 1 - 504 Sum[DivisorSigma[5, n]*q^(2 n), {n, 1, terms}]; s = 17/24 E4[q]^3 + 7/24 E6[q]^2 + O[q]^(3 terms); Partition[ CoefficientList[s, q], 2][[All, 1]][[1 ;; terms]] (* Jean-François Alcover, Jul 06 2017 *)

Formula

This series is the q-expansion of (17*E_4(z)^3 + 7*E_6(z)^2)/24. - Daniel D. Briggs, Nov 25 2011

A008698 Theta series of Niemeier lattice of type A_7^2 D_5^2.

Original entry on oeis.org

1, 192, 191952, 16821504, 397751376, 4630308480, 34416494784, 187486720512, 814895994960, 2975529668544, 9486529043040, 27053048565504, 70486165778112, 169930984401024, 384163663508352
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

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

Links

Crossrefs

Cf. A004009, A013973.
Cf. A008688 - A008697, A008699 - A008704.

Programs

  • Mathematica
    terms = 15; E4[q_] := 1 + 240 Sum[DivisorSigma[3, n]*q^(2 n), {n, 1, terms}]; E6[q_] := 1 - 504 Sum[DivisorSigma[5, n]*q^(2 n), {n, 1, terms}]; s = 25/36 E4[q]^3 + 11/36 E6[q]^2 + O[q]^(3 terms); Partition[ CoefficientList[s, q], 2][[All, 1]][[1 ;; terms]] (* Jean-François Alcover, Jul 06 2017 *)

Formula

This series is the q-expansion of (25*E_4(z)^3 + 11*E_6(z)^2)/36. - Daniel D. Briggs, Nov 25 2011
Showing 1-10 of 15 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.