%I A112301
%S A112301 1,2,0,0,2,0,0,0,1,4,0,0,2,0,0,0,2,2,0,0,0,0,0,0,3,4,0,0,2,0,0,0,0,4,0,
%T A112301 0,2,0,0,0,2,0,0,0,2,0,0,0,1,6,0,0,2,0,0,0,0,4,0,0,2,0,0,0,4,0,0,0,0,0,
%U A112301 0,0,2,4,0,0,0,0,0,0,1,4,0,0,4,0,0,0,2,4,0,0,0,0,0,0,2,2,0,0,2,0,0,0,0
%V A112301 1,-2,0,0,2,0,0,0,1,-4,0,0,2,0,0,0,2,-2,0,0,0,0,0,0,3,-4,0,0,2,0,0,0,0,
               -4,0,0,2,0,0,0,
%W A112301 2,0,0,0,2,0,0,0,1,-6,0,0,2,0,0,0,0,-4,0,0,2,0,0,0,4,0,0,0,0,0,0,0,2,-4,
               0,0,0,0,0,0,1,
%X A112301 -4,0,0,4,0,0,0,2,-4,0,0,0,0,0,0,2,-2,0,0,2,0,0,0,0
%N A112301 Expansion of (eta(q) * eta(q^16))^2 / (eta(q^2) * eta(q^8)) in powers 
               of q.
%F A112301 Euler transform of period 16 sequence [ -2, -1, -2, -1, -2, -1, -2, 0, 
               -2, -1, -2, -1, -2, -1, -2, -2, ...].
%F A112301 Multiplicative with a(2)=-2, a(2^e) = 0 if e>1, a(p^e) = e+1 if p == 
               1 (mod 4), a(p^e) = (1+(-1)^e)/2 if p == 3 (mod 4).
%F A112301 G.f.: Product_{k>0} (1-x^k)^2(1+x^(8k))^2(1+x^(2k))(1+x^(4k)).
%F A112301 Expansion of q * phi(-q) * psi(q^8) in powers of q where phi(), psi() 
               are Ramanujan theta functions.
%F A112301 Expansion of (phi(-q^2)^2 - phi(-q)^2) / 4 in powers of q where phi() 
               is a Ramanujan theta function.
%F A112301 Moebius transform is period 16 sequence [ 1, -3, -1, 2, 1, 3, -1, 0, 
               1, -3, -1, -2, 1, 3, -1, 0, ...].
%F A112301 G.f. is a period 1 Fourier series that satisfies f(-1 / (16 t)) = 4 (t/
               i) f(t) where q = exp(2 pi i t).
%F A112301 a(4n) = a(4n+3) = a(8n+6) = 0. a(8n+2) = -2 * a(4n+1).
%F A112301 G.f.: Sum_{k>0} kronecker(-4, k) * x^k * (1 - x^k)^2 / (1 - x^(4*k)).
%e A112301 q - 2*q^2 + 2*q^5 + q^9 - 4*q^10 + 2*q^13 + 2*q^17 - 2*q^18 + 3*q^25 
               - ...
%o A112301 (PARI) {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( (eta(x+A)*eta(x^16+A))^2/
               (eta(x^2+A)*eta(x^8+A)), n))}
%o A112301 (PARI) {a(n) = if( n>0 & (n+1)%4\2, (n%2*3 - 2) * sumdiv( n/gcd(n,2), 
               d, (-1)^(d\2)))}
%Y A112301 -(-1)^n * A134013(n) = a(n). A008441(n) = a(4*n+1). A113407(n) = a(8*n+1). 
               2 * A053692(n) = a(8*n+5).
%Y A112301 Sequence in context: A107497 A000095 A034949 this_sequence A134013 A136521 
               A066448
%Y A112301 Adjacent sequences: A112298 A112299 A112300 this_sequence A112302 A112303 
               A112304
%K A112301 sign,mult
%O A112301 1,2
%A A112301 Michael Somos, Sep 02 2005, Oct 02 2007