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.

A028595 Duplicate of A002653.

Original entry on oeis.org

1, 6, 24, 56, 114, 168, 280, 294, 444, 390, 840, 636, 1176, 1176, 1512
Offset: 0

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Author

Keywords

A002706 Theta series of 6-dimensional lattice A_6^(2) (other names for this lattice or the corresponding quadratic form are LAMBDA_{3,lambda}, P_6^(5), phi_6, F_14).

Original entry on oeis.org

1, 0, 42, 56, 84, 168, 280, 336, 462, 336, 840, 672, 1176, 1176, 1386, 1008, 1848, 2016, 2058, 2520, 3528, 2408, 3108, 2688, 4760, 3024, 5880, 4592, 6468, 4704, 5040, 6720, 6930, 6832, 10080, 7224, 7812, 7392, 12600, 7056, 14280, 11760, 12040, 9408
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

In Elkies 1999 the g.f. is denoted by theta_L. - Michael Somos, Nov 09 2014

Examples

			G.f. = 1 + 42*q^2 + 56*q^3 + 84*q^4 + 168*q^5 + 280*q^6 + 336*q^7 + 462*q^8 + ...

References

  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, Intro. to 3rd ed.
  • N. Elkies, The Klein quartic in number theory, pp. 51-101 of S. Levy, ed., The Eightfold Way, Cambridge Univ. Press, 1999. MR1722413 (2001a:11103). See page 72.

Links

Crossrefs

Cf. A002653, A002656.

Programs

  • Magma
    A := Basis( ModularForms( Gamma1(7), 3), 44); A[1] + 42*A[3] + 56*A[4] + 84*A[5] + 168*A[6] + 280*A[7];  /* Michael Somos, Nov 09 2014 */
  • Mathematica
    s = (EllipticTheta[3, 0, q] *EllipticTheta[3, 0, q^7] + EllipticTheta[2, 0, q]*EllipticTheta[2, 0, q^7])^3 - 6q*(QPochhammer[q] *QPochhammer[q^7])^3 + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 04 2015, from first formula *)
  • PARI
    {a(n) = local(A, t1, t2, t3); if( n<1, n==0, A = x * O(x^n); t1 = x * (eta(x + A) * eta(x^7 + A))^3; t2 = sum(k=1, (sqrtint(4*n + 1)  + 1)\2, 2 * x^(k*k - k), A); t3 = sum(k=1, sqrtint(n), 2 * x^(k*k), 1 + A); A = x * O(x^(n\7)); polcoeff( (t3 * subst(t3 + A, x, x^7) + x^2 * t2 * subst(t2 + A, x, x^7))^3 - 6*t1, n))}; /* Michael Somos, Jun 03 2005 */
  • Sage
    A = ModularForms( Gamma1(7), 3, prec=25) . basis(); (-21*A[0] + 4*A[1] + 21*A[2] + 105*A[3] + 224*A[4] + 441*A[5] + 672*A[6])/4 # Michael Somos, May 25 2014

Formula

a(n) = A002653(n) - 6*A002656(n).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) is a homogeneous degree 6 polynomial with 28 terms. - Michael Somos, Jun 03 2005

A028596 Expansion of (theta_3(z)*theta_3(7z) + theta_2(z)*theta_2(7z))^4.

Original entry on oeis.org

1, 8, 40, 128, 328, 656, 1216, 1864, 2856, 3560, 5392, 6368, 9856, 10640, 17000, 16832, 22600, 23760, 32776, 32576, 43792, 52864, 57568, 58560, 78528, 76024, 94864, 98432, 137864, 116720, 152512, 143040, 179240, 179072, 212112, 237328, 265768, 242352, 296704, 295232
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A002652, A002653, A028594.

Programs

  • PARI
    a(n) = polcoeff((1 + 2*x*Ser(qfrep([2, 1; 1, 4], n, 1)))^4, n); \\ Jinyuan Wang, Feb 21 2020

Extensions

More terms from Jinyuan Wang, Feb 21 2020
Showing 1-3 of 3 results.

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

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