%I A030187
%S A030187 1,1,2,1,0,2,1,1,1,0,0,2,4,1,0,1,6,1,2,0,2,0,0,2,5,4,4,1,6,0,4,1,0,6,0,
%T A030187 1,2,2,8,0,6,2,8,0,0,0,12,2,1,5,12,4,6,4,0,1,4,6,6,0,8,4,1,1,0,0,4,6,0,
%U A030187 0,0,1,2,2,10,2,0,8,8,0,11,6,6,2,0,8,12,0,6,0,4,0,8,12,0,2,10,1,0,5,0
%V A030187 1,-1,-2,1,0,2,1,-1,1,0,0,-2,-4,-1,0,1,6,-1,2,0,-2,0,0,2,-5,4,4,1,-6,0,
               -4,-1,0,-6,0,1,
%W A030187 2,-2,8,0,6,2,8,0,0,0,-12,-2,1,5,-12,-4,6,-4,0,-1,-4,6,-6,0,8,4,1,1,0,
               0,-4,6,0,0,0,-1,
%X A030187 2,-2,10,2,0,-8,8,0,-11,-6,-6,-2,0,-8,12,0,-6,0,-4,0,8,12,0,2,-10,-1,0,
               -5,0
%N A030187 Expansion of eta(q)*eta(q^2)*eta(q^7)*eta(q^14) in powers of q.
%D A030187 M. Koike, Matheiu group M24 and modular forms, Nagoya Math. J., 99 (1985), 
               147-157. MR0805086 (87e:11060)
%F A030187 Euler transform of period 14 sequence [ -1, -2, -1, -2, -1, -2, -2, -2, 
               -1, -2, -1, -2, -1, -4, ...]. - Michael Somos Aug 13 2006
%F A030187 a(n) is multiplicative with a(2^e) = (-1)^e, a(7^e) = 1, otherwise a(p^e) 
               = a(p)a(p^(e-1))-p*a(p^(e-2)) where a(p) = p minus number of points 
               of elliptic curve modulo p . - Michael Somos Aug 13 2006
%F A030187 G.f.: x Product_{k>0} (1-x^k)(1-x^(2k))(1-x^(7k))(1-x^(14k)).
%F A030187 Coefficients of L-series for elliptic curve "14a4": y^2 +x*y +y= x^3 
               -x or y^2 +x*y -y= x^3 . - Michael Somos Feb 19 2007
%F A030187 G.f. A(x) satisfies 0= f(A(x), A(x^2), A(x^4)) where f(u, v, w)= v^4 
               -u*w* (u+2*v)* (v+2*w) . - Michael Somos Feb 19 2007
%F A030187 Associated with permutations in Mathieu group M24 of shape (14)(7)(2)(1).
%F A030187 G.f. is Fourier series of a weight 2 level 14 modular form. f(-1/ (14 
               t)) = 14 (t/i)^2 f(t) where q = exp(2 pi i t).
%o A030187 (PARI) {a(n)=if(n<1, 0, ellak(ellinit([ -1, 0, -1, -1, 0]), n))} /* Michael 
               Somos Aug 13 2006 */
%o A030187 (PARI) {a(n)= local(A, p, e, x, y, a0, a1); if(n<1, 0, A=factor(n); prod(k=1, 
               matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, (-1)^e, if(p==7, 
               1, a0=1; a1=y=-sum(x=0, p-1, kronecker(4*x^3+x^2-2*x+1, p)); for(i=2, 
               e, x=y*a1-p*a0; a0=a1; a1=x); a1)))))} /* Michael Somos Aug 13 2006 
               */
%o A030187 (PARI) {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff(eta(x+A)*eta(x^2+A)*eta(x^7+A)*eta(x^14+A), 
               n))}
%Y A030187 Sequence in context: A130182 A024361 A135486 this_sequence A117278 A140082 
               A025852
%Y A030187 Adjacent sequences: A030184 A030185 A030186 this_sequence A030188 A030189 
               A030190
%K A030187 sign,mult
%O A030187 1,3
%A A030187 N. J. A. Sloane (njas(AT)research.att.com).