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 1-2 of 2 results.

A008409 Theta series of 16-dimensional Barnes-Wall lattice.

Original entry on oeis.org

1, 0, 4320, 61440, 522720, 2211840, 8960640, 23224320, 67154400, 135168000, 319809600, 550195200, 1147643520, 1771683840, 3371915520, 4826603520, 8593797600, 11585617920, 19590534240, 25239859200, 40979580480, 50877235200
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			1 + 4320*q^4 + 61440*q^6 + 522720*q^8 + 2211840*q^10 + 8960640*q^12 + ...

References

  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 130, p. 131 Equation (132).

Links

Crossrefs

A008774(2*n) = a(n).
Cf. A004009, A319307.

Programs

  • Mathematica
    f[q_] := 1/2*(EllipticTheta[2, 0, q]^16 + EllipticTheta[3, 0, q]^16 + EllipticTheta[4, 0, q]^16 + 30*EllipticTheta[2, 0, q]^8*EllipticTheta[3, 0, q]^8); Series[f[q], {q, 0, 21}] // CoefficientList[#, q]& (* Jean-François Alcover, May 15 2013 *)
  • PARI
    {a(n) = local(A1, A2) ; if( n<0, 0, A1 = eta(x + x * O(x^n))^8; A2 = eta(x^2 + x * O(x^n))^8; polcoeff( (A1^6 + 32 * x * A1^3 * A2^3 + 4096 * x^2 * A2^6) / ( A1 * A2 )^2, n))} /* Michael Somos, Nov 29 2007 */

Formula

Expansion of ( theta_2(q)^16 + theta_3(q)^16 + theta_4(q)^16 + 30 * theta_2(q)^8 * theta_3(q)^8 ) / 2 in powers of q. - [Conway and Sloane]
Expansion of E_4(q^2)^2 + (E_4(q) - E_4(q^2))^2 / 15 in powers of q. - Michael Somos, Nov 29 2007
Expansion of ( eta(q)^48 + 32 * eta(q)^24 * eta(q^2)^24 + 4096 * eta(q^2)^48 ) / ( eta(q) * eta(q^2) )^16 in powers of q. - Michael Somos, Nov 29 2007
G.f. is Fourier series of a weight 8 level 2 modular form. f(-1 / (2 t)) = 16 (t/i)^8 f(t) where q = exp(2 Pi i t). - Michael Somos, Nov 29 2007

A135828 Expansion of psi(x^2)^8 * (psi(x)^8 + psi(-x)^8) / 2 in powers of x^2 where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, 36, 378, 2200, 8955, 28836, 78558, 188568, 410805, 828080, 1564686, 2804976, 4809370, 7927380, 12643560, 19594632, 29568204, 43626708, 63094550, 89501040, 124916931, 171803652, 232822908, 311683680, 412601490, 539849556, 699657642, 898801400, 1143680535
Offset: 0

Views

Author

Michael Somos, Nov 29 2007

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + 36*x + 378*x^2 + 2200*x^3 + 8955*x^4 + 28836*x^6 + 78558*x^7 + ...
G.f. = q^3 + 36*q^5 + 378*q^7 + 2200*q^9 + 8955*q^11 + 28836*q^13 + 78558*q^15 + ...

Links

Crossrefs

Cf. A007331, A008774, A045823.

Programs

  • Magma
    Basis( ModularForms( Gamma1(4), 8), 60)[4]; /* Michael Somos, Oct 15 2015 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x]^8 (EllipticTheta[ 2, 0, x^(1/2)]^8 + EllipticTheta[ 2, Pi/4, x^(1/2)]^8 16) / 131072, {x, 0, 2 n + 3}]; (* Michael Somos, Oct 15 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, n *= 2; A = x * O(x^n); polcoeff( ( eta(x^2 + A)^24 + eta(x + A)^16 * eta(x^4 + A)^8 ) / ( 2 * eta(x + A)^8 * eta(x^2 + A)^16 / eta(x^4 + A)^16 ), n))};

Formula

Expansion of q^(-3) * ( eta(q^2)^24 + eta(q)^16 * eta(q^4)^8 ) / ( 2 * eta(q)^8 * eta(q^2)^16 / eta(q^4)^16 ) in powers of q^2.
7680 * a(n) = A008774(2*n + 3).
Convolution of A007331 and A045823.
Showing 1-2 of 2 results.

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

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