A023918 Theta series of A*_6 lattice.

1, 0, 0, 14, 0, 42, 70, 42, 0, 0, 210, 0, 294, 294, 210, 0, 0, 504, 0, 630, 882, 350, 0, 0, 1190, 0, 1470, 1148, 882, 0, 0, 1680, 0, 1708, 2520, 1050, 0, 0, 3150, 0, 3570, 2940, 1750, 0, 0, 3066, 0, 3864, 4774, 2100, 0, 0, 6174, 0, 5740, 5124, 3570, 0, 0, 6090

Offset

0,4

Comments

Positions of nonzero entries seem to be A047328. - Andrey Zabolotskiy, Nov 10 2021

References

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 114.

Links

Table of n, a(n) for n=0..59.

S. Ahlgren, The sixth, eighth, ninth and tenth powers of Ramanujan's theta function, Proc. Amer. Math. Soc., 128 (1999), 1333-1338; F_7(q).

G. Nebe and N. J. A. Sloane, Home page for this lattice

Formula

Expansion of f(-x)^7 / f(-x^7) + 7 * x * f(-x)^3 * f(-x^7)^3 + 7 * x^2 * f(-x^7)^7 / f(-x) in powers of x where f() is a Ramanujan theta function. - Michael Somos, Jan 29 2011

a(7*n) = A008446(n). a(7*n + 1) = a(7*n + 2) = a(7*n + 4) = 0. - Michael Somos, Jan 29 2011

Example

1 + 14*x^3 + 42*x^5 + 70*x^6 + 42*x^7 + 210*x^10 + 294*x^12 + 294*x^13 + ...

Mathematica

a[n_] := Module[{A, A7}, A = x*O[x]^n; A7 = QPochhammer[x^7 + A]; A = QPochhammer[x + A]; SeriesCoefficient[A^7 / A7 + 7 * x * (A * A7)^3 + 7 * x^2 * A7^7 / A, {x, 0, n}]]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Nov 05 2015, adapted from Michael Somos's PARI script *)

Prog

(PARI) {a(n) = local(A, A7); if( n<0, 0, A = x * O(x^n); A7 = eta(x^7 + A); A = eta(x + A); polcoeff( A^7 / A7 + 7 * x * (A * A7)^3 + 7 * x^2 * A7^7 / A, n))}; /* Michael Somos, Jan 29 2011 */

Crossrefs

Cf. A008446.

Cf. theta series of lattices A*_0, A*_1, A*_2, A*_3, A*_4...: A000007, A000122, A004016, A004013, A023916, A023917, this sequence, A023919-A023936.

Cf. A047328.

Sequence in context: A008423 A240252 A317977 * A062785 A161384 A337104

Adjacent sequences: A023915 A023916 A023917 * A023919 A023920 A023921

Keyword

nonn

Author

N. J. A. Sloane

Status

approved