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 1-5 of 5 results.

A008704 Theta series of Niemeier lattice of type A_1^24.

Original entry on oeis.org

1, 48, 195408, 16785216, 397963344, 4629612960, 34417365696, 187489131648, 814883829840, 2975546033136, 9486545735520, 27052971581376, 70486219194048, 169931067595296, 384163605641088
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 - A008703.

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 = 11/18 E4[q]^3 + 7/18 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 (11*E_4(z)^3 + 7*E_6(z)^2)/18. - Daniel D. Briggs, Nov 26 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

A008701 Theta series of Niemeier lattice of type A_4^6.

Original entry on oeis.org

1, 120, 193680, 16803360, 397857360, 4629960720, 34416930240, 187487926080, 814889912400, 2975537850840, 9486537389280, 27053010073440, 70486192486080, 169931025998160, 384163634574720
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 - A008700, A008702, 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 = 47/72 E4[q]^3 + 25/72 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 (47*E_4(z)^3 + 25*E_6(z)^2)/72. - Daniel D. Briggs, Nov 25 2011

A055765 Jacobi form of weight 12 and index 1 for Niemeier lattice of type A_2^12.

Original entry on oeis.org

1, 0, 0, 2, 66, 0, 0, 32208, 130284, 0, 0, 3676662, 9177368, 0, 0, 95643312, 188222238, 0, 0, 1143073734, 1959783480, 0, 0, 8506602000, 13292902392, 0, 0, 45762314600, 67073793600, 0, 0, 195390284832, 272567987604, 0, 0
Offset: 0

Views

Author

Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jul 12 2000

Keywords

References

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

Crossrefs

A008703, A055747, A003785.

Formula

E_8*E_{4, 1}-54*phi_12.
G.f.: b(z) - 54*c(z) where b(z) is the g.f. for A055747 and c(z) is the g.f. for A003785. - Sean A. Irvine, Apr 05 2022
Showing 1-5 of 5 results.

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

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