A035016 Fourier coefficients of E_{0,4}.

1, -16, 112, -448, 1136, -2016, 3136, -5504, 9328, -12112, 14112, -21312, 31808, -35168, 38528, -56448, 74864, -78624, 84784, -109760, 143136, -154112, 149184, -194688, 261184, -252016, 246176, -327040, 390784, -390240, 395136, -476672, 599152, -596736

Offset

0,2

Comments

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).

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

References

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 77, Eq. (31.61).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)

Richard E. Borcherds, Automorphic forms with singularities on Grassmannians, arXiv:alg-geom/9609022, 1996-1997; Invent. Math. 132 (1998), 491-562.

B. Brent, Quadratic Minima and Modular Forms, Experimental Mathematics, v.7 no.3, 257-274.

Formula

a(0)=1; for n>0, a(n) = 16*sum_{0<d|n}(-1)^d d^3.

G.f.: Product_{n>=1} ((1-q^n)/(1+q^n))^8 [Fine]

Expansion of phi(-q)^8 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Jun 15 2014

Expansion of eta(q)^16 / eta(q^2)^8 in powers of q.

Euler transform of period 2 sequence [ -16, -8, ...]. - Michael Somos, Apr 10 2005

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

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

a(n) = (-1)^n * A000143(n).

Convolution square of A096727. - Michael Somos, Jun 15 2014

Example

G.f. = 1 - 16*q + 112*q^2 - 448*q^3 + 1136*q^4 - 2016*q^5 + 3136*q^6 - 5504*q^7 + ...

Maple

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

Mathematica

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 *)
QP = QPochhammer; s = QP[q]^16/QP[q^2]^8 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015 *)

Prog

(PARI) {a(n) = if( n<1, n==0, 16 * sumdiv( n, d, (-1)^d * d^3))};
(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))};
(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 */
(Sage) A = ModularForms( Gamma0(4), 4, prec=34) . basis(); A[0] - 16*A[1] + 112*A[2]; # Michael Somos, Jun 15 2014
(Python)
from sympy import divisors
def a(n): return 1 if n==0 else 16*sum((-1)**d*d**3 for d in divisors(n))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 24 2017
(Julia) # JacobiTheta4 is defined in A002448.
A035016List(len) = JacobiTheta4(len, 8)
A035016List(34) |> println # Peter Luschny, Mar 12 2018

Crossrefs

Cf. A000143, A096727.

Sequence in context: A371114 A279425 A144449 * A000143 A258546 A205964

Adjacent sequences: A035013 A035014 A035015 * A035017 A035018 A035019

Keyword

sign,easy,nice

Author

Barry Brent (barryb(AT)primenet.com)

Status

approved