0,2

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).

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

T. D. Noe, Table of n, a(n) for n=0..1000

Borcherds, Richard E., Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998), 491-562.

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

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 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+uv(u-2v+16w)-16uw^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 g() is g.f. for A007331. - Michael Somos Jan 11 2009

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

(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 */

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

Sequence in context: A107908 A144449 A000143 this_sequence A081194 A121148 A091031

Adjacent sequences: A035013 A035014 A035015 this_sequence A035017 A035018 A035019

sign,easy,nice

Barry Brent (barryb(AT)primenet.com)