A003781 Expansion of theta series of {E_7}* lattice in powers of q^(1/2).

1, 0, 0, 56, 126, 0, 0, 576, 756, 0, 0, 1512, 2072, 0, 0, 4032, 4158, 0, 0, 5544, 7560, 0, 0, 12096, 11592, 0, 0, 13664, 16704, 0, 0, 24192, 24948, 0, 0, 27216, 31878, 0, 0, 44352, 39816, 0, 0, 41832, 55944, 0, 0, 72576, 66584, 0, 0, 67536, 76104, 0, 0, 100800

Offset

0,4

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.

M. Eichler and D. Zagier, The Theory of Jacobi Forms, Birkhauser, 1985, p. 141.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Theta series is given on page 125 of Conway and Sloane.

Can be determined from A023919 (A*_7): [1] A003781(4n)=A023919(16n) [2] A003781(4n+3)=A023919(16n+12). Let A_7+[1] be the generator of A*_7/A_7, then these correspond to [1]A004008=theta(E_7)=theta(A_7)+theta(A_7+[4]), [2]A005931=theta(E_7+[1])=theta(A_7+[2])+theta(A_7+[6]) - Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), May 03 2000

Expansion of phi(q)^3 * (phi(q)^4 + 7 * phi(-q)^4) / 8 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Aug 27 2013

G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 2^(13/2) (t / i)^(7/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A004008. - Michael Somos, Aug 27 2013

a(4*n + 1) = a(4*n + 2) = 0. - Michael Somos, Jun 11 2007

a(4*n) = A004008(n), a(4*n + 3) = A005931(n). - Michael Somos, Jun 11 2007.

Example

G.f. = 1 + 56*x^3 + 126*x^4 + 576*x^7 + 756*x^8 + 1512*x^11 + 2072*x^12 + ...
G.f. = 1 + 56*q^(3/2) + 126*q^2 + 576*q^(7/2) + 756*q^4 + 1512*q^(11/2) + ...

Mathematica

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

Prog

(PARI) {a(n) = local(A, B, m); n++; m=n%4; n\=4; if( n<0 || m>1, 0, A = sum(k=1, sqrtint(n), 2*x^k^2, 1 + x * O(x^n)); B = subst(A, x, -x); polcoeff( if(m==0, (A^4 - B^4) * (8*A^4 - B^4) / 2 / sum(k=0, sqrtint( 4*n + 1)\2, x^(k^2 + k), x * O(x^n)), 8*A^7 - 7*A^3 * subst(A, x, -x)^4 ), n))}; /* Michael Somos, Jun 11 2007 */
(PARI) {a(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))}; /* Michael Somos, Aug 27 2013 */
(Magma) Basis( ModularForms( Gamma0(4), 7/2), 19) [1] ; /* Michael Somos, Jun 10 2014 */

Crossrefs

Cf. A004008, A005931, A030443, A038723.

Sequence in context: A044624 A157330 A038849 * A286980 A254463 A030443

Adjacent sequences: A003778 A003779 A003780 * A003782 A003783 A003784

Keyword

nonn,nice

Author

N. J. A. Sloane

Status

approved