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 1-4 of 4 results.

A047807 Duplicate of A008700.

Original entry on oeis.org

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

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Author

Keywords

Comments

Original title: Theta series of Niemeier lattice of type A_5^4*D_4.

Crossrefs

Equal to the theta series of D_4^6, A008700.

Programs

  • Mathematica
    terms = 16; 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 = 2/3 E4[q]^3 + 1/3 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 2/3 E_4(z)^3 + 1/3 E_6(z)^2. Cf. A004009, A013973. - Daniel D. Briggs, Nov 25 2011

Extensions

More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jul 05 2000

A008699 Theta series of Niemeier lattice of type A_6^4.

Original entry on oeis.org

1, 168, 192528, 16815456, 397786704, 4630192560, 34416639936, 187487122368, 814893967440, 2975532395976, 9486531825120, 27053035734816, 70486174680768, 169930998266736, 384163653863808
Offset: 0

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Author

Keywords

References

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

Crossrefs

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 = 49/72 E4[q]^3 + 23/72 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 (49*E_4(z)^3 + 23*E_6(z)^2)/72. - Daniel D. Briggs, Nov 25 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

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Author

Keywords

References

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

Crossrefs

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

A055762 Jacobi form of weight 12 and index 1 for Niemeier lattice of type A_5^4 D_4 or D_4^6.

Original entry on oeis.org

1, 0, 0, 8, 126, 0, 0, 31680, 129492, 0, 0, 3684312, 9181784, 0, 0, 95595072, 188204958, 0, 0, 1143217944, 1959861960, 0, 0, 8506517184, 13292577672, 0, 0, 45761541536, 67074486336, 0, 0, 195392621952, 272568215700, 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

Formula

E_8*E_{4, 1}-48*phi_12.
G.f.: b(z) - 48*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 1-4 of 4 results.