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A035016 Fourier coefficients of E_{0,4}.

%I A035016 #52 Apr 09 2020 11:35:06
%S A035016 1,-16,112,-448,1136,-2016,3136,-5504,9328,-12112,14112,-21312,31808,
%T A035016 -35168,38528,-56448,74864,-78624,84784,-109760,143136,-154112,149184,
%U A035016 -194688,261184,-252016,246176,-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).
%C A035016 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A035016 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 77, Eq. (31.61).
%H A035016 Seiichi Manyama, <a href="/A035016/b035016.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)
%H A035016 Richard E. Borcherds, <a href="http://arXiv.org/abs/alg-geom/9609022">Automorphic forms with singularities on Grassmannians</a>, arXiv:alg-geom/9609022, 1996-1997; Invent. Math. 132 (1998), 491-562.
%H A035016 B. Brent, <a href="http://projecteuclid.org/euclid.em/1047674207">Quadratic Minima and Modular Forms</a>, Experimental Mathematics, v.7 no.3, 257-274.
%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 phi(-q)^8 in powers of q where phi() is a Ramanujan theta function. - _Michael Somos_, Jun 15 2014
%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 + u*v * (u - 2*v + 16*w) - 16 * u*w^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 q = exp(2 Pi i t) and g() is the g.f. for A007331. - _Michael Somos_, Jan 11 2009
%F A035016 a(n) = (-1)^n * A000143(n).
%F A035016 Convolution square of A096727. - _Michael Somos_, Jun 15 2014
%e A035016 G.f. = 1 - 16*q + 112*q^2 - 448*q^3 + 1136*q^4 - 2016*q^5 + 3136*q^6 - 5504*q^7 + ...
%p A035016 a_list := proc(len) series(JacobiTheta4(0,x)^8,x,len+1); seq(coeff(%,x,j),j=0..len) end: a_list(33); # _Peter Luschny_, Mar 14 2017
%t A035016 a[0] = 1; a[n_] := 16*Sum[(-1)^d*d^3, {d, Divisors[n]}]; Table[a[n], {n, 0, 33}] (* _Jean-François Alcover_, Feb 06 2012, after Pari *)
%t A035016 QP = QPochhammer; s = QP[q]^16/QP[q^2]^8 + O[q]^40; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 25 2015 *)
%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 */
%o A035016 (Sage) A = ModularForms( Gamma0(4), 4, prec=34) . basis(); A[0] - 16*A[1] + 112*A[2]; # _Michael Somos_, Jun 15 2014
%o A035016 (Python)
%o A035016 from sympy import divisors
%o A035016 def a(n): return 1 if n==0 else 16*sum((-1)**d*d**3 for d in divisors(n))
%o A035016 print([a(n) for n in range(101)]) # _Indranil Ghosh_, Jun 24 2017
%o A035016 (Julia) # JacobiTheta4 is defined in A002448.
%o A035016 A035016List(len) = JacobiTheta4(len, 8)
%o A035016 A035016List(34) |> println # _Peter Luschny_, Mar 12 2018
%Y A035016 Cf. A000143, A096727.
%K A035016 sign,easy,nice
%O A035016 0,2
%A A035016 Barry Brent (barryb(AT)primenet.com)