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 1-8 of 8 results.

A004033 Theta series of lattice A_2 tensor E_8 (dimension 16, det. 6561, min. norm 4). Also theta series of Eisenstein version of E_8 lattice.

Original entry on oeis.org

1, 0, 720, 13440, 97200, 455040, 1714320, 4821120, 12380400, 29043840, 58980960, 114076800, 219310320, 367338240, 621878400, 1037727360, 1583679600, 2401816320, 3747180240, 5232470400, 7551983520, 10938261120, 14715224640, 19930775040, 28073386800, 35727920640
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Also theta series of 16-dimensional lattice (SL(2,9) Y SL(2,9)).(C2 x C2). - John Cannon, Jan 10 2007
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + 720*x^2 + 13440*x^3 + 97200*x^4 + 455040*x^5 + 1714320*x^6 + 4821120*x^7 + ...
G.f. = 1 + 720*q^4 + 13440*q^6 + 97200*q^8 + 455040*q^10 + 1714320*q^12 + 4821120*q^14 + ...

References

  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A004016, A007332, A037150.

Programs

  • Magma
    // Definition for lattice (SL(2,9) Y SL(2,9)).(C2 x C2), from John Cannon
LatticeWithBasis(16, \[ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ], MatrixRing(IntegerRing(), 16) ! \[
4, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 4, 2, 1, 1, 1, 2,
1, -1, 1, 1, 0, 1, 0, 1, 1, 2, 2, 4, 0, 1, 2, 2, 1, 1, 1, 2, 1, 0, 1,
1, 2, 1, 1, 0, 4, 1, 1, 1, 2, 0, 2, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 4,
1, 1, 1, 1, 0, 0, 0, 1, 2, 2, 2, 2, 1, 2, 1, 1, 4, 1, 1, 2, 1, 2, 2,
0, 2, 1, 2, 1, 2, 2, 1, 1, 1, 4, 1, 0, 2, 2, 1, 1, 0, 1, 1, 1, 1, 1,
2, 1, 1, 1, 4, 1, 2, 2, 2, 1, 1, 1, 0, 1, -1, 1, 0, 1, 2, 0, 1, 4, 1,
1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 0, 1, 2, 2, 1, 4, 2, 2, 1, 0, 0, -1, 2,
1, 2, 1, 0, 2, 2, 2, 1, 2, 4, 2, 0, 0, 1, 1, 1, 0, 1, 1, 0, 2, 1, 2,
2, 2, 2, 4, 1, 0, -1, 1, 1, 1, 0, 2, 1, 0, 1, 1, 1, 1, 0, 1, 4, 1, 1,
1, 2, 0, 1, 1, 2, 2, 0, 1, 2, 0, 0, 0, 1, 4, 2, 1, 2, 1, 1, 1, 2, 1,
1, 1, 1, 0, 1, -1, 1, 2, 4, 1, 1, 1, 2, 0, 2, 2, 1, 0, 1, -1, 1, 1, 1,
1, 1, 4 ])
  • Magma
    // Definition for lattice A_2 tensor E_8, from John Cannon
A := Lattice("A", 2);
B := Lattice("E", 8);
L := TensorProduct(A, B);
T := ThetaSeries(L, 16);
  • Magma
    A := Basis( ModularForms( Gamma0(3), 8), 26); A[1] + 720*A[3]; /* Michael Somos, Feb 01 2017 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ With[ {a1 = (QPochhammer[ x]^3 + 9 x QPochhammer[ x^9]^3) / QPochhammer[ x^3]}, a1^2 (a1^6 - 48 x QPochhammer[ x]^6 QPochhammer[ x^3]^6)], {x, 0, n}]; (* Michael Somos, Feb 01 2017 *)
  • PARI
    {a(n) = if( n<0, 0, my(A, a1); A = x * O(x^n); a1 = (eta(x + A)^3 + 9*x * eta(x^9 + A)^3) / eta(x^3 + A); polcoeff( a1^2 * (a1^6 - 48*x * eta(x + A)^6 * eta(x^3 + A)^6), n))}; /* Michael Somos, Feb 01 2017 */

Formula

Theta series is x^8-48*x^2*y, x = phi_0(z) (see A004016), y = Delta_12(z) (see A007332) in the notation of SPLAG, Chap. 4. See A037150 for Maple code.
Expansion of a(x)^2 * (a(x)^6 - 48*x * f(-x)^6 * f(-x^3)^6) in powers of x where a() is a cubic AGM theta function and f() is a Ramanujan theta function. - Michael Somos, Feb 01 2017
G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 81 (t/i)^8 f(t) where q = exp(2 Pi i t). - Michael Somos, Feb 01 2017

A037146 Numerators of Fourier coefficients of Eisenstein series of degree 2 and weight 10 when evaluated at Gram(A_2)*z.

Original entry on oeis.org

1, 0, -792, 227244864, -9944907192, 919209728448, 34981193422296, 642291149636928, 8088505738922664, 75871526303850816, 561429626003520912, 3435029183941210944, 17943112657234300008, 82098865517562748800, 335623489808760863424, 1245022670592143885376
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • Helmut Klingen, Introductory Lectures on Siegel Modular Forms, p. 123.

Links

Crossrefs

Cf. A037147.

Formula

G.f.: x^20 - 120*x^14*y + 1728*x^8*y^2 + (1330587648/43867)*x^2*y^3 where x = phi_0(z) and y = Delta_12(z). See A037150 for definitions of these quantities. - N. J. A. Sloane, Dec 12 2020

Extensions

More terms from Sean A. Irvine, Dec 12 2020

A037147 Denominators of Fourier coefficients of Eisenstein series of degree 2 and weight 10 when evaluated at Gram(A_2)*z.

Original entry on oeis.org

1, 1, 1, 43867, 43867, 43867, 43867, 43867, 43867, 43867, 43867, 43867, 43867, 43867, 43867, 43867, 43867, 43867, 43867, 43867, 43867, 43867, 43867, 43867, 43867, 43867, 43867, 43867, 43867, 43867, 43867, 43867, 43867, 43867, 43867, 43867, 43867, 43867, 43867
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • Helmut Klingen, Introductory Lectures on Siegel Modular Forms, p. 123.

Links

Crossrefs

Cf. A037146, A037150.

Formula

x^20 - 120*x^14*y + 1728*x^8*y^2 + (1330587648/43867)*x^2*y^3, x = phi_0(z), y = Delta_12(z). (See A037150 for explanation and Maple code. - N. J. A. Sloane, Dec 12 2020)

Extensions

More terms from Sean A. Irvine, Dec 13 2020

A037148 Numerators of Fourier coefficients of Eisenstein series of degree 2 and weight 12 when evaluated at Gram(A_2)*z.

Original entry on oeis.org

1, 0, 196560, 22266840960, 32657336384400, 488671648133760, 53439465983183280, 1517285377810500480, 33146268593842731600, 495735471753920012160, 5599704694679105905440, 50142366540860643504000, 370980488114849951891280, 2338140291564951182050560
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • Helmut Klingen, Introductory Lectures on Siegel Modular Forms, p. 123.

Links

Crossrefs

Cf. A037149, A037150.

Formula

x^24 - 144*x^18*y + (3480192/691)*x^12*y^2 - (2037901234176/53678953)*x^6*y^3 + (21009383424000/53678953)*y^4, x = phi_0(z), y = Delta_12(z). (see A037150 for definitions and Maple code - N. J. A. Sloane, Dec 12 2020)

Extensions

More terms from Sean A. Irvine, Dec 13 2020

A037149 Denominators of Fourier coefficients of Eisenstein series of degree 2 and weight 12 when evaluated at Gram(A_2)*z.

Original entry on oeis.org

1, 1, 691, 53678953, 53678953, 53678953, 53678953, 53678953, 53678953, 53678953, 53678953, 53678953, 53678953, 53678953, 53678953, 53678953, 53678953, 53678953, 53678953, 53678953, 53678953, 53678953, 53678953, 53678953, 53678953, 53678953, 53678953, 53678953
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • Helmut Klingen, Introductory Lectures on Siegel Modular Forms, p. 123.

Links

Crossrefs

Cf. A037148, A037150.

Formula

x^24 - 144*x^18*y + (3480192/691)*x^12*y^2 - (2037901234176/53678953)*x^6*y^3 + (21009383424000/53678953)*y^4, x = phi_0(z), y = Delta_12(z). See A037150 for definitions and Maple code.

Extensions

More terms from Sean A. Irvine, Dec 13 2020

A037190 Theta series of lattice A_2 tensor D12+ (dimension 24, min norm 4).

Original entry on oeis.org

1, 0, 792, 16704, 573480, 6289344, 47367288, 256779072, 1116635400, 4092791616, 12993723216, 37059359040, 96956352072, 232789209984, 526238331840, 1128121363008, 2286165374280, 4451600668032, 8386296787224, 15130668404160, 26614226534832, 45684867221568
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Formula

G.f.: phi_0(z)^12 - 72*phi_0(z)^6*Delta_12(z) + 576*Delta_12(z)^2 where phi_0(z) is the theta series of the A_2 lattice given in A004016, and Delta_12 is the 12-dimensional cusp form given in A007332. - Sean A. Irvine, Dec 16 2020

Extensions

More terms from Sean A. Irvine, Dec 16 2020

A037191 Theta series of lattice A_2 tensor E_7^2+ (dimension 28, min norm 4).

Original entry on oeis.org

1, 0, 756, 17472, 700812, 13487040, 143699220, 1061439552, 6038839548, 27868766016, 109814205816, 378998996544, 1173180111852, 3325312497024, 8715279904416, 21337957069632, 49450420013148, 108741984430464, 228329201738484, 461698406495424, 899506102392072
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Formula

G.f.: phi_0(z)^14 - 84*phi_0(z)^8*Delta_12(z) + 1008*phi_0(z)^2*Delta_12(z)^2 where phi_0(z) is the theta series of the A_2 lattice given in A004016, and Delta_12 is the 12-dimensional cusp form given in A007332. - Sean A. Irvine, Dec 16 2020

Extensions

More terms from Sean A. Irvine, Dec 16 2020

A037212 Theta series of lattice A_2 tensor A_15+ (dimension 30, det 3^15, min. norm 4).

Original entry on oeis.org

1, 0, 720, 17640, 726570, 17236728, 223847280, 1928224440, 12500672580, 65073916440, 284229347328, 1080456070680, 3650778821970, 11191823745120, 31612753519200, 83017908015624, 204808695842610, 479043914067360, 1065816689968080, 2271147404056200
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A004016, A007332.

Formula

G.f.: phi_0(z)^15 - 90*phi_0(z)^9*Delta_12(z) + 1260*phi_0(z)^3*Delta_12(z)^2 where phi_0(z) is the theta series of the A_2 lattice given in A004016, and Delta_12 is the 12-dimensional cusp form given in A007332. - Sean A. Irvine, Dec 17 2020

Extensions

More terms from Sean A. Irvine, Dec 17 2020
Showing 1-8 of 8 results.

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

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