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

%I A003781 #25 Feb 16 2025 08:32:27
%S A003781 1,0,0,56,126,0,0,576,756,0,0,1512,2072,0,0,4032,4158,0,0,5544,7560,0,
%T A003781 0,12096,11592,0,0,13664,16704,0,0,24192,24948,0,0,27216,31878,0,0,
%U A003781 44352,39816,0,0,41832,55944,0,0,72576,66584,0,0,67536,76104,0,0,100800
%N A003781 Expansion of theta series of {E_7}* lattice in powers of q^(1/2).
%C A003781 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A003781 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 125.
%D A003781 M. Eichler and D. Zagier, The Theory of Jacobi Forms, Birkhauser, 1985, p. 141.
%H A003781 G. C. Greubel, <a href="/A003781/b003781.txt">Table of n, a(n) for n = 0..1000</a>
%H A003781 N. Elkies and B. H. Gross, <a href="http://dx.doi.org/10.2140/pjm.1997.181.147">Embeddings into the integral octonions, Olga Taussky-Todd: in memoriam</a>, Pacific J. Math. 1997, Special Issue, 147-158.
%H A003781 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A003781 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A003781 Theta series is given on page 125 of Conway and Sloane.
%F A003781 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
%F A003781 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
%F A003781 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
%F A003781 a(4*n + 1) = a(4*n + 2) = 0. - _Michael Somos_, Jun 11 2007
%F A003781 a(4*n) = A004008(n), a(4*n + 3) = A005931(n). - _Michael Somos_, Jun 11 2007.
%e A003781 G.f. = 1 + 56*x^3 + 126*x^4 + 576*x^7 + 756*x^8 + 1512*x^11 + 2072*x^12 + ...
%e A003781 G.f. = 1 + 56*q^(3/2) + 126*q^2 + 576*q^(7/2) + 756*q^4 + 1512*q^(11/2) + ...
%t A003781 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 *)
%o A003781 (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 */
%o A003781 (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 */
%o A003781 (Magma) Basis( ModularForms( Gamma0(4), 7/2), 19) [1] ; /* _Michael Somos_, Jun 10 2014 */
%Y A003781 Cf. A004008, A005931, A030443, A038723.
%K A003781 nonn,nice
%O A003781 0,4
%A A003781 _N. J. A. Sloane_