%I A107760
%S A107760 1,3,3,3,3,0,3,6,3,3,0,0,3,6,6,0,3,0,3,6,0,6,0,0,3,3,6,3,6,0,0,6,3,0,0,
%T A107760 0,3,6,6,6,0,0,6,6,0,0,0,0,3,9,3,0,6,0,3,0,6,6,0,0,0,6,6,6,3,0,0,6,0,0,
%U A107760 0,0,3,6,6,3,6,0,6,6,0,3,0,0,6,0,6,0,0,0,0,12,0,6,0,0,3,6,9,0,3,0,0,6,
               6
%N A107760 Expansion of eta(q^3)eta(q^2)^6/(eta(q)^3 eta(q^6)^2) in powers of q.
%D A107760 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. 
               Soc., 1988; p. 80, Eq. (32.42).
%F A107760 Euler transform of period 6 sequence [3, -3, 2, -3, 3, -2, ...].
%F A107760 G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)= +v^3 
               +u^2*w +4*v*w^2 -4*v^2*w -2*u*v*w.
%F A107760 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-u2)*(u1-u2-u3+u6) -3*u6*(u2-u6).
%F A107760 Expansion of psi(q)^3/psi(q^3) in powers of q where psi() is a Ramanujan 
               theta function.
%F A107760 Expansion of (a(q) + a(q^2)) / 2 = b(q^2)^2 / b(q) in powers of q where 
               a(), b() are cubic AGM functions. - Michael Somos, Aug 30 2008
%F A107760 Moebius transform is period 6 sequence [ 3, 0, 0, 0, -3, 0, ...]. - Michael 
               Somos Aug 11 2009
%F A107760 a(n) = 3 * b(n) unless n=0 and b(n) is multiplicative with b(p^e) = 1 
               if p=2 or p=3; b(p^e) = 1+e if p == 1 (mod 6); b(p^e) = (1+(-1)^e)/
               2 if p == 5 (mod 6). - Michael Somos Aug 11 2009
%F A107760 G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (27/
               4)^(1/2) (t/i) g(t) where q = exp(2 pi i t) and g() is g.f. for A123330. 
               - Michael Somos Aug 11 2009
%F A107760 G.f.: (Product_{k>0} (1 - x^(2*k)) / (1 - x^(2*k-1)))^3 / (Product_{k>
               0} (1 - x^(6*k)) / (1 - x^(6*k-3))). - Michael Somos Aug 11 2009
%e A107760 1 + 3*q + 3*q^2 + 3*q^3 + 3*q^4 + 3*q^6 + 6*q^7 + 3*q^8 + 3*q^9 + ... 
               - Michael Somos Aug 11 2009
%o A107760 (PARI) a(n)=if(n<1,n==0,3*direuler(p=2,n,1/(1-X)/(1-kronecker(-12,p)*X))[n])
%o A107760 (PARI) {a(n)=local(A); if(n<0,0, A=x*O(x^n); polcoeff( eta(x^3+A)*eta(x^2+A)^6/
               eta(x^6+A)^2/eta(x+A)^3, n))}
%o A107760 (PARI) {a(n)=if(n<1, n==0, 3*sumdiv(n, d, kronecker(-12, d)))}
%Y A107760 a(n)=3*A035178(n), if n>0.
%Y A107760 Sequence in context: A033700 A122916 A132973 this_sequence A138070 A081334 
               A106694
%Y A107760 Adjacent sequences: A107757 A107758 A107759 this_sequence A107761 A107762 
               A107763
%K A107760 nonn
%O A107760 0,2
%A A107760 Michael Somos, May 24 2005