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 A110399 Expansion of (theta_3(q)theta_3(q^7)-1)/2 in powers of q. 0

%I

%S 1,0,0,1,0,0,1,2,1,0,2,0,0,0,0,3,0,0,0,0,0,0,2,0,1,0,0,1,2,0,0,4,0,0,

%T 0,1,2,0,0,0,0,0,2,2,0,0,0,0,1,0,0,0,2,0,0,2,0,0,0,0,0,0,1,5,0,0,2,0,

%U 0,0,2,2,0,0,0,0,2,0,2,0,1,0,0,0,0,0,0,4,0,0,0,2,0,0,0,0,0,0,2,1,0,0,0,0,0

%N Expansion of (theta_3(q)theta_3(q^7)-1)/2 in powers of q.

%D B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 302 Entry 17(ii)

%F a(n) is multiplicative and a(2^e) = |e-1|, a(7^e)= 1, a(p^e) = e+1 if p == 1, 2, 4 (mod 7), a(p^e) = (1+(-1)^e)/2 if p == 3, 5, 6 (mod 7).

%F G.f.: Sum_{k>0}, kronecker(-7, k) x^k/(1-(-x)^k).

%F G.f.: (theta_3(q)theta_3(q^7)-1)/2 where theta_3(q)=1+2(q+q^4+q^9+...).

%o (PARI) {a(n)=local(x); if(n<1, 0, x=valuation(n,2); abs(x-1)*sumdiv(n/2^x,d,kronecker(-28,d)))}

%o (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, e-1, if(p==7, 1, if(kronecker(-7,p)==-1, (1+(-1)^e)/2, e+1))))))}

%o (PARI) {a(n)=local(A); if(n<1, 0, A=x*O(x^n); polcoeff( (eta(x+A)^-2*eta(x^2+A)^5 *eta(x^4+A)^-2*eta(x^7+A)^-2*eta(x^14+A)^5 *eta(x^28+A)^-2-1)/2, n))}

%Y A033719(n)=2 a(n) if n>0. A035162(2n+1)=A035182(2n+1)=a(2n+1).

%K nonn,mult

%O 1,8

%A Michael Somos, Oct 22 2005

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