A004008 Expansion of theta series of E_7 lattice in powers of q^2.

1, 126, 756, 2072, 4158, 7560, 11592, 16704, 24948, 31878, 39816, 55944, 66584, 76104, 99792, 116928, 133182, 160272, 177660, 205128, 249480, 265104, 281736, 350784, 382536, 390726, 470232, 505568, 532800, 615384, 640080, 701568, 799092, 809424, 853776

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

Examples

			G.f. = 1 + 126*x + 756*x^2 + 2072*x^3 + 4158*x^4 + 7560*x^5 + 11592*x^6 + ...
G.f. = 1 + 126*q^2 + 756*q^4 + 2072*q^6 + 4158*q^8 + 7560*q^10 + 11592*q^12 + ...
		

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 125. Equation (112)
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• G. Nebe and N. J. A. Sloane, Home page for this lattice
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A003781, A005875, A228746.

Programs

Magma

A := Basis( ModularForms( Gamma0(4), 7/2), 50); A[1] + 126*A[2]; /* Michael Somos, Jun 09 2014 */

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^3 ( 8 EllipticTheta[ 3, 0, q]^4 - 7 EllipticTheta[ 4, 0, q]^4), {q, 0, n}]; (* Michael Somos, Aug 27 2013 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^3 ( EllipticTheta[ 3, 0, q]^4 + 7 EllipticTheta[ 2, 0, q]^4), {q, 0, n}]; (* Michael Somos, Apr 21 2015 *)

PARI

{a(n) = my(A); if( n<0, 0, A = sum(k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n)); polcoeff( A^3 * (8 * A^4 - 7 * subst(A, x, -x)^4), n))}; /* Michael Somos, Oct 24 2006 */

PARI

{a(n) = my(G); if( n<1, n==0, G = [2, -1, 0, 0, 0, 0, 0; -1, 2, -1, 0, 0, 0, 0; 0, -1, 2, -1, 0, 0, 0; 0, 0, -1, 2, -1, 0, -1; 0, 0, 0, -1, 2, -1, 0; 0, 0, 0, 0, -1, 2, 0; 0, 0, 0, -1, 0, 0, 2]; 2 * qfrep( G, n, 1)[n])}; /* Michael Somos, Jun 11 2007 */

Formula

Expansion of phi(q)^3 * (phi(q)^4 + 7 * 16 * q * psi(q^2)^4) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Oct 24 2006

G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 2^(1/2) (t / i)^(7/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A003781. - Michael Somos, Aug 27 2013

Convolution of A005875 and A228746. - Michael Somos, Apr 21 2015