Search: id:A122855
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%I A122855
%S A122855 1,1,0,1,1,1,0,0,2,1,0,0,1,0,0,1,3,2,0,2,1,0,0,2,2,1,0,1,0,0,0,2,4,0,0,
%T A122855 0,1,0,0,0,2,0,0,0,0,1,0,2,3,1,0,2,0,2,0,0,0,2,0,0,1,2,0,0,5,0,0,0,2,2,
%U A122855 0,0,2,0,0,1,2,0,0,2,3,1,0,2,0,2,0,0,0,0,0,0,2,2,0,2,4,0,0,0,1,0,0,0,0
%N A122855 Expansion of (phi(q^3)phi(q^5)+phi(q)phi(q^15))/2 in powers of q where 
               phi(q) is a Ramanujan theta function.
%H A122855 A. Berkovich and H. Yesilyurt, <a href="http://arXiv.org/abs/math.NT/
               0611300">Ramanujan's identities and representation of integers by 
               certain binary and quaternary quadratic forms</a>
%F A122855 Expansion of (eta(q^2)^2*eta(q^6)eta(q^10)eta(q^30)^2)/(eta(q)eta(q^4)eta(q^15)eta(q^60)) 
               in powers of q.
%F A122855 a(n) is multiplicative with a(2^e) = |e-1|, a(3^e)=a(5^e)=1, a(p^e) = 
               e+1 if p == 1, 2, 4, 8 (mod 15), a(p^e) = (1+(-1)^e)/2 if p == 7, 
               11, 13, 14 (mod 15).
%F A122855 Euler transform of period 60 sequence [ 1, -1, 1, 0, 1, -2, 1, 0, 1, 
               -2, 1, -1, 1, -1, 2, 0, 1, -2, 1, -1, 1, -1, 1, -1, 1, -1, 1, 0, 
               1, -4, 1, 0, 1, -1, 1, -1, 1, -1, 1, -1, 1, -2, 1, 0, 2, -1, 1, -1, 
               1, -2, 1, 0, 1, -2, 1, 0, 1, -1, 1, -2, ...].
%F A122855 Moebius transform is period 60 sequence [ 1, -1, 0, 1, 0, 0, -1, 1, 0, 
               0, -1, 0, -1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, -1, -1, 0, 
               1, 1, 0, -1, 0, 0, -1, -1, 0, 0, -1, 0, -1, -1, 0, -1, 1, 0, 1, 0, 
               0, -1, 1, 0, 0, -1, 0, 1, -1, 0, ...].
%F A122855 a(15n+7)=a(15n+11)=a(15n+13)=a(15n+14)=0. a(3n)=a(5n)=a(n).
%F A122855 G.f.: 1+Sum_{k>0} kronecker(-15,k) x^k/(1-(-x)^k).
%o A122855 (PARI) {a(n)=if(n<1, n==0, sumdiv(n, d, kronecker(-15,d)*(-1)^(d%4==2)))}
%o A122855 (PARI) {a(n)= local(A, p, e); if(n<1, n==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(p%15==2^valuation(p%15,
               2), e+1, 1-e%2))))))}
%o A122855 (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^2*eta(x^6+A)*eta(x^10+A)*eta(x^30+A)^2/ 
               (eta(x+A)*eta(x^4+A)*eta(x^15+A)*eta(x^60+A)), n))}
%Y A122855 A035175(n)=a(4n).
%Y A122855 Sequence in context: A086017 A000161 A060398 this_sequence A140727 A140728 
               A130068
%Y A122855 Adjacent sequences: A122852 A122853 A122854 this_sequence A122856 A122857 
               A122858
%K A122855 nonn,mult
%O A122855 0,9
%A A122855 Michael Somos, Sep 14 2006

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