0,3

"For Gamma, it is known that any modular form is a weighted homogeneous polynomial in Theta_Z, which has weight 1/2 and the modular form delta_8(t) := e^{pi i tau} prod_{m=1..infty} ((1-e^{ pi i m tau}) (1+e^{2 pi i m tau}))^8 = e^{ pi i tau} - 8 e^{2 pi i m tau} +28 e^{3 pi i m tau} -64 e^{4 pi i m tau} +126 e^{5 pi i m tau} ... of weight 4." [Elkies, p. 1242]

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 187.

N. D. Elkies. Lattices, Linear Codes and Invariants, Part I. Amer. Math. Soc., 47 (No. 10, Nov. 2000), 1238-1245, see p. 1242.

F. Hirzebruch et al., Manifolds and Modular Forms, Vieweg 1994 p 133.

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

Expansion of (eta(q)eta(q^4)/eta(q^2))^8 in powers of q. - Michael Somos, Jul 16 2004

Euler transform of period 4 sequence [ -8,0,-8,-8,...]. - Michael Somos, Jul 16 2004

G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)= +u^4*w*v +16*u^3*w*v^2 +16*u^2*w^2*v^2 +256*u^3*w^3 +256*u^3*w^2*v +4096*u^2*w^3*v +4096*u*w^4*v +4096*u*w^3*v^2 -u^2*v^4 -16*u^2*w*v^3 -256*u*w^2*v^3 -256*w^2*v^4 . - Michael Somos May 31 2005

G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6)=u1^4*u6^4 +u1^3*u2*u3^3*u6 +2*u1*u2^3*u3*u6^3 -u2^4*u3^4.

Expansion of q * psi(-q)^8 in powers of q where psi() is a Ramanujan theta function. - Michael Somos Mar 20 2008

a(n) is multiplicative with a(2^e) = -8^e if e>0, a(p^e) = ((p^3)^(e+1) - 1) / (p^3 - 1). - Michael Somos Mar 20 2008

G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 16 (t/i)^4 f(t) where q = exp(2 pi i t).

G.f.: x * (Product_{k>0} (1 - x^(2*k-1)) * (1 - x^(4*k)))^8.

q - 8*q^2 + 28*q^3 - 64*q^4 + 126*q^5 - 224*q^6 + 344*q^7 ...

q*product((1-q^(2*k-1))^8*(1-q^(4*k))^8, k=1..75);

(PARI) a(n)=local(A); if(n<1, 0, n--; A=x^n*O(x); polcoeff((eta(x+A)/eta(x^2+A)*eta(x^4+A))^8, n)) /* Michael Somos, Jul 16 2004 */

(PARI) a(n)=local(A); if(n<1, 0, n--; A=x^n*O(x); polcoeff((prod(k=1, n, (1-(k%4==0)*x^k)*(1-(k%2==1)*x^k), 1+A))^8, n)) /* Michael Somos, Jul 16 2004 */

(PARI) a(n)=if(n<1, 0, -(-1)^n*sumdiv(n, d, (n/d%2)*d^3)) /* Michael Somos May 31 2005 */

a(n)=-(-1)^n*A007331(n).

Sequence in context: A045850 A033580 A007331 this_sequence A101127 A007259 A134747

Adjacent sequences: A002405 A002406 A002407 this_sequence A002409 A002410 A002411

sign,nice,easy,mult

N. J. A. Sloane (njas(AT)research.att.com) and Mira Bernstein (mira(AT)math.berkeley.edu)