A030443 Nonzero coefficients in theta series of {E_7}* lattice.

1, 56, 126, 576, 756, 1512, 2072, 4032, 4158, 5544, 7560, 12096, 11592, 13664, 16704, 24192, 24948, 27216, 31878, 44352, 39816, 41832, 55944, 72576, 66584, 67536, 76104, 100800, 99792, 101304, 116928, 145728, 133182, 126504, 160272, 205632, 177660, 176456, 205128, 249984, 249480, 234360

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

In the Eichler and Zagier reference this is e_4(A014601(n)), n >= 0, (p. 141), where e_4 is obtained from e_{4,1}(n,r), eq. (7), p. 22, depending only on 4*n-r^2 >= 0 (for integers n and r), i.e. on A014601(n), n >= 0 (with a new notation for n). - Wolfdieter Lang, Jan 08 2016

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 125.
• M. Eichler and D. Zagier, The Theory of Jacobi Forms, Birkhäuser, 1985, p. 141.

Links

• N. Elkies and B. H. Gross, Embeddings into the integral octonions, Olga Taussky-Todd: in memoriam, Pacific J. Math. 1997, Special Issue, 147-158.

Crossrefs

Cf. A003781.

Programs

PARI

f(n) = local(A); if( n<0, 0, A = sum(k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n)); polcoeff( A^3 * (A^4 + 7 * subst(A, x, -x)^4) / 8, n)); \\ A003781
lista(nn) = select(x->(x>0), vector(nn, k, f(k-1))); \\ Michel Marcus, Nov 11 2023

Extensions

More terms from Michel Marcus, Nov 11 2023