A004008 Expansion of theta series of E_7 lattice in powers of q^2. (Formerly M5388)

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

0,2

Comments

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

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

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

Example

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 + ...

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 *)

Prog

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

Crossrefs

Cf. A003781, A005875, A228746.

Sequence in context: A186817 A107658 A181254 * A292981 A126170 A324709

Adjacent sequences: A004005 A004006 A004007 * A004009 A004010 A004011

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved