%I A123484
%S A123484 1,2,1,0,0,2,2,0,1,0,0,0,2,4,0,0,0,2,2,0,2,0,0,0,1,4,1,0,0,0,2,0,0,0,0,
%T A123484 0,2,4,2,0,0,4,2,0,0,0,0,0,3,2,0,0,0,2,0,0,2,0,0,0,2,4,2,0,0,0,2,0,0,0,
%U A123484 0,0,2,4,1,0,0,4,2,0,1,0,0,0,0,4,0,0,0,0,4,0,2,0,0,0,2,6,0,0,0,0,2,0,0
%V A123484 1,-2,1,0,0,-2,2,0,1,0,0,0,2,-4,0,0,0,-2,2,0,2,0,0,0,1,-4,1,0,0,0,2,0,
               0,0,0,0,2,-4,2,0,
%W A123484 0,-4,2,0,0,0,0,0,3,-2,0,0,0,-2,0,0,2,0,0,0,2,-4,2,0,0,0,2,0,0,0,0,0,2,
               -4,1,0,0,-4,2,0,
%X A123484 1,0,0,0,0,-4,0,0,0,0,4,0,2,0,0,0,2,-6,0,0,0,0,2,0,0
%N A123484 Expansion of eta(q)^2 * eta(q^6)^4 * eta(q^8) * eta(q^24) / (eta(q^2) 
               * eta(q^3) * eta(q^12))^2 in powers of q.
%C A123484 Expansion of (a(q) - 2 * a(q^2) - a(q^4) + 2*a(q^8)) / 6 in powers of 
               q where a() is a cubic AGM function.
%F A123484 Euler transform of period 24 sequence [ -2, 0, 0, 0, -2, -2, -2, -1, 
               0, 0, -2, 0, -2, 0, 0, -1, -2, -2, -2, 0, 0, 0, -2, -2, ...].
%F A123484 Moebius transform is period 24 sequence [ 1, -3, 0, 2, -1, 0, 1, 0, 0, 
               3, -1, 0, 1, -3, 0, 0, -1, 0, 1, -2, 0, 3, -1, 0, ...].
%F A123484 a(n) is multiplicative with a(2) = -2, a(2^e) = 0 if e>1, a(3^e) = 1, 
               a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1+(-1)^e)/2 if p == 5 (mod 
               6).
%F A123484 G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 12^(1/
               2) (t/i) g(t) where q = exp(2 pi i t) and g() is g.f. for A136748.
%F A123484 G.f.: x * Product_{k>0} (1 -x^(6*k)) * (1 - x^k + x^(2*k))^2 * (1 - x^(8*k)) 
               * (1 + x^(12*k)) / (1 + x^(6*k)).
%F A123484 a(4*n) = a(6*n + 4) = a(6*n + 5) = 0. a(3*n) = a(n).
%e A123484 q - 2*q^2 + q^3 - 2*q^6 + 2*q^7 + q^9 + 2*q^13 - 4*q^14 - 2*q^18 + ...
%o A123484 (PARI) {a(n)=if(n<1, 0, sumdiv(n, d, if(d%2, 1, d/2%2*-2)*kronecker(-12, 
               n/d)))}
%o A123484 (PARI) {a(n)=local(A, p, e); 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, -2*(e<2), if(p==3, 1, if(p%6==1, 
               e+1, !(e%2)))))))}
%o A123484 (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x 
               + A)^2 * eta(x^6 + A)^4 * eta(x^8 + A) * eta(x^24 + A) / (eta(x^2 
               + A) * eta(x^3 + A) * eta(x^12 + A))^2, n))}
%Y A123484 A033762(n) = a(2*n+1). A112604(n) = a(4*n+1). -2 * A033762(n) = a(4*n+2). 
               A112605(n) = a(4*n+3). A097195(n) = a(6*n+1). A112606(n) = a(8*n+1). 
               -2 * A112604(n) = a(8*n+2). A112608(n) = a(8*n+3). 2 * A112607(n) 
               = a(8*n+5). -2 * A112605(n) = a(8*n+6). 2 * A112609(n) = a(8*n+7).
%Y A123484 A123884(n) = a(12*n+1). 2 * A121361(n) = a(12*n+7). A131961(n) = a(24*n+1). 
               2 * A131962(n) = a(24*n+7). A112608(n) = a(24*n+9). 2 * A131963(n) 
               = a(24*n+13). 2 * A131964(n) = a(24*n+19).
%Y A123484 Sequence in context: A161528 A136176 A103344 this_sequence A008626 A058626 
               A122856
%Y A123484 Adjacent sequences: A123481 A123482 A123483 this_sequence A123485 A123486 
               A123487
%K A123484 sign,mult
%O A123484 1,2
%A A123484 Michael Somos, Sep 28 2006, Apr 04 2008