%I A002408
%S A002408 0,1,8,28,64,126,224,344,512,757,1008,1332,1792,2198,2752,3528,4096,4914,
               6056,
%T A002408 6860,8064,9632,10656,12168,14336,15751,17584,20440,22016,24390,28224,
               29792,
%U A002408 32768,37296,39312,43344,48448,50654,54880,61544,64512,68922
%V A002408 0,1,-8,28,-64,126,-224,344,-512,757,-1008,1332,-1792,2198,-2752,3528,
               -4096,4914,-6056,
%W A002408 6860,-8064,9632,-10656,12168,-14336,15751,-17584,20440,-22016,24390,-28224,
               29792,
%X A002408 -32768,37296,-39312,43344,-48448,50654,-54880,61544,-64512,68922
%N A002408 Expansion of 8-dimensional cusp form.
%C A002408 "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]
%D A002408 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", 
               Springer-Verlag, p. 187.
%D A002408 N. D. Elkies. Lattices, Linear Codes and Invariants, Part I. Amer. Math. 
               Soc., 47 (No. 10, Nov. 2000), 1238-1245, see p. 1242.
%D A002408 F. Hirzebruch et al., Manifolds and Modular Forms, Vieweg 1994 p 133.
%H A002408 T. D. Noe, <a href="b002408.txt">Table of n, a(n) for n=0..1000</a>
%F A002408 Expansion of (eta(q)eta(q^4)/eta(q^2))^8 in powers of q. - Michael Somos, 
               Jul 16 2004
%F A002408 Euler transform of period 4 sequence [ -8,0,-8,-8,...]. - Michael Somos, 
               Jul 16 2004
%F A002408 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
%F A002408 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.
%F A002408 Expansion of q * psi(-q)^8 in powers of q where psi() is a Ramanujan 
               theta function. - Michael Somos Mar 20 2008
%F A002408 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
%F A002408 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).
%F A002408 G.f.: x * (Product_{k>0} (1 - x^(2*k-1)) * (1 - x^(4*k)))^8.
%e A002408 q - 8*q^2 + 28*q^3 - 64*q^4 + 126*q^5 - 224*q^6 + 344*q^7 ...
%p A002408 q*product((1-q^(2*k-1))^8*(1-q^(4*k))^8, k=1..75);
%o A002408 (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 */
%o A002408 (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 */
%o A002408 (PARI) a(n)=if(n<1, 0, -(-1)^n*sumdiv(n,d,(n/d%2)*d^3)) /* Michael Somos 
               May 31 2005 */
%Y A002408 a(n)=-(-1)^n*A007331(n).
%Y A002408 Sequence in context: A045850 A033580 A007331 this_sequence A101127 A007259 
               A134747
%Y A002408 Adjacent sequences: A002405 A002406 A002407 this_sequence A002409 A002410 
               A002411
%K A002408 sign,nice,easy,mult
%O A002408 0,3
%A A002408 N. J. A. Sloane (njas(AT)research.att.com) and Mira Bernstein (mira(AT)math.berkeley.edu)