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
Examples
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) + ...
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
Programs
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Magma
Basis( ModularForms( Gamma0(4), 7/2), 19) [1] ; /* Michael Somos, Jun 10 2014 */
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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 *) -
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 */
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
Comments