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 1-3 of 3 results.

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

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

A055756 Jacobi form of weight 12 and index 1 for Niemeier lattice of type A_12^2.

Original entry on oeis.org

1, 0, 0, 22, 266, 0, 0, 30448, 127644, 0, 0, 3702162, 9192088, 0, 0, 95482512, 188164638, 0, 0, 1143554434, 1960045080, 0, 0, 8506319280, 13291819992, 0, 0, 45759737720, 67076102720, 0, 0, 195398075232, 272568747924, 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

A008694, A055747, A003785.

Formula

E_8*E_{4, 1}-34*phi_12.
G.f.: b(z) + 32*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 1-3 of 3 results.

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

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