%I A035016
%S A035016 1,16,112,448,1136,2016,3136,5504,9328,12112,14112,21312,31808,35168,
%T A035016 38528,56448,74864,78624,84784,109760,143136,154112,149184,194688,
%U A035016 261184,252016,246176,327040,390784,390240,395136,476672,599152,596736
%V A035016 1,-16,112,-448,1136,-2016,3136,-5504,9328,-12112,14112,-21312,31808,-35168,
               38528,
%W A035016 -56448,74864,-78624,84784,-109760,143136,-154112,149184,-194688,261184,
               -252016,246176,
%X A035016 -327040,390784,-390240,395136,-476672,599152,-596736
%N A035016 Fourier coefficients of E_{0,4}.
%C A035016 E_{0,4} is unique normalized entire modular form of weight 4 for \Gamma_0(2) 
               with a zero at zero. Also |a(n)| matches expansion of theta_3(z)^8 
               (A000143).
%D A035016 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. 
               Soc., 1988; p. 77, Eq. (31.61).
%H A035016 T. D. Noe, <a href="b035016.txt">Table of n, a(n) for n=0..1000</a>
%H A035016 Borcherds, Richard E., <a href="http://arXiv.org/abs/alg-geom/9609022">
               Automorphic forms with singularities on Grassmannians</a>, Invent. 
               Math. 132 (1998), 491-562.
%H A035016 B. Brent, <a href="http://www.expmath.org/expmath/volumes/7/7.html">Quadratic 
               Minima and Modular Forms, Experimental Mathematics, v.7 no.3, 257-274.</
               a>
%F A035016 a(0)=1; for n>0, a(n) = 16*sum_{0<d|n}(-1)^d d^3.
%F A035016 G.f.: Product_{n>=1} ((1-q^n)/(1+q^n))^8 [Fine]
%F A035016 Expansion of eta(q)^16/eta(q^2)^8 in powers of q.
%F A035016 Euler transform of period 2 sequence [ -16, -8, ...]. - Michael Somos, 
               Apr 10 2005
%F A035016 G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)=v^3+uv(u-2v+16w)-16uw^2. 
               - Michael Somos Apr 10 2005
%F A035016 G.f. is a period 1 Fourier series which satisfies f(-1 / (2 t)) = 256 
               (t / i)^4 g(t) where g() is g.f. for A007331. - Michael Somos Jan 
               11 2009
%e A035016 1 - 16*q + 112*q^2 - 448*q^3 + 1136*q^4 - 2016*q^5 + 3136*q^6 - 5504*q^7 
               + ...
%o A035016 (PARI) a(n)=if(n<1,n==0,16*sumdiv(n,d,(-1)^d*d^3))
%o A035016 (PARI) {a(n)=if(n<0,0, polcoeff( prod(k=1,n,(1-x^k)/(1+x^k), 1+x*O(x^n))^8,
               n))}
%o A035016 (PARI) {a(n) = local(A); if( n<0, 0, A = x^n * O(x); polcoeff( (eta(x 
               + A)^2 / eta(x^2 + A))^8, n))} /* Michael Somos Jan 11 2009 */
%Y A035016 (-1)^n * A000143(n) = a(n).
%Y A035016 Sequence in context: A107908 A144449 A000143 this_sequence A081194 A121148 
               A091031
%Y A035016 Adjacent sequences: A035013 A035014 A035015 this_sequence A035017 A035018 
               A035019
%K A035016 sign,easy,nice
%O A035016 0,2
%A A035016 Barry Brent (barryb(AT)primenet.com)