%I A030188
%S A030188 1,1,2,0,1,4,2,2,2,4,0,8,1,1,6,8,4,0,6,2,6,4,2,0,7,2,2,8,4,4,2,0,4,4,8,
%T A030188 8,10,1,0,8,1,4,4,6,6,0,8,8,2,4,18,16,0,12,2,6,18,16,2,0,5,6,12,8,4,4,
               0,
%U A030188 2,6,12,0,8,12,7,14,16,2,16,2,2,0,12,8,24,9,4,6,0,4,12,6,2,12,8,0,0
%V A030188 1,-1,-2,0,1,4,-2,2,2,-4,0,-8,-1,-1,6,8,-4,0,6,2,-6,4,-2,0,-7,-2,-2,-8,
               4,4,-2,0,4,-4,8,
%W A030188 8,10,1,0,-8,1,-4,-4,-6,-6,0,-8,8,2,4,-18,16,0,-12,-2,-6,18,16,-2,0,5,
               6,12,-8,-4,-4,0,
%X A030188 2,-6,-12,0,-8,-12,7,14,-16,2,-16,-2,2,0,12,8,24,-9,-4,6,0,-4,12,6,2,-12,
               8,0,0
%N A030188 Expansion of q^(-1)eta(q^2)*eta(q^4)*eta(q^6)*eta(q^12) in powers of 
               q^2.
%D A030188 M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
%D A030188 J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson 
               Education, Inc, 2006, p. 415. Exer. 47.3.
%H A030188 W. Stein, <a href="http://modular.fas.harvard.edu:8080/mfd/newform.html?space=[24,
               2,[]]&number=1&search=24">Modular Forms Database</a>.
%F A030188 Euler transform of period 6 sequence [ -1, -2, -2, -2, -1, -4, ...]. 
               - Michael Somos Apr 2 2005
%F A030188 Given g.f. A(x), then B(x)=x*A(x)^2 satisfies 0=f(B(x), B(x^2), B(x^4)) 
               where f(u, v, w)=u^2vw+4uv^2w+16uvw^2+4u^2w^2-v^4. - Michael Somos 
               Apr 2 2005
%F A030188 a(n)=b(2n+1) where b(n) is multiplicative and b(2^e)=0^e, b(3^e)=(-1)^e, 
               b(p^e) =b(p)*b(p^(e-1)) -p*b(p^(e-2)) otherwise where b(p)= p - number 
               of solutions to y^2=x^3-x^2-4*x+4 modulo p. - Michael Somos Aug 13 
               2006
%F A030188 G.f.: Product_{k>0} (1-x^k)(1-x^(2k))(1-x^(3k))(1-x^(6k)).
%F A030188 Newform number 1 of degree 1 in Full modular forms space of level 24, 
               weight 2 and trivial character.
%F A030188 G.f. is Fourier series of a weight 2 level 24 modular form. f(-1/ (24 
               t)) = 24 (t/i)^2 f(t) where q = exp(2 pi i t). - Michael Somos Jun 
               08 2007
%F A030188 L-series for elliptic curve "24a1": y^2 = x^3 - x^2 - 4*x + 4. - Michael 
               Somos Apr 2 2005
%e A030188 q - q^3 - 2*q^5 + q^9 + 4*q^11 - 2*q^13 + 2*q^15 + 2*q^17 - 4*q^19 + 
               ...
%o A030188 (PARI) {a(n)=local(A); if(n<0,0, A=x*O(x^n); polcoeff( eta(x+A)*eta(x^2+A)*eta(x^3+A)*eta(x^6+A), 
               n))} /* Michael Somos Apr 2 2005 */
%o A030188 (PARI) a(n)=if(n<0,0, n=2*n+1; ellak(ellinit([0,-1,0,-4,4]),n)) /* Michael 
               Somos Apr 2 2005 */
%o A030188 (PARI) {a(n)=local(A, p, e, x, y, a0, a1); if(n<0, 0, n=2*n+1; A=factor(n); 
               prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, 0, if(p==3, 
               (-1)^e, a0=1; a1=y=-sum(x=0, p-1, kronecker(x^3-x^2-4*x+4, p)); for(i=2, 
               e, x=y*a1-p*a0; a0=a1; a1=x); a1)))))} /* Michael Somos Aug 13 2006 
               */
%Y A030188 Sequence in context: A124915 A158239 A159819 this_sequence A160648 A124912 
               A138752
%Y A030188 Adjacent sequences: A030185 A030186 A030187 this_sequence A030189 A030190 
               A030191
%K A030188 sign
%O A030188 0,3
%A A030188 N. J. A. Sloane (njas(AT)research.att.com).