%I A113417
%S A113417 1,2,4,8,7,10,12,8,18,18,16,24,21,20,28,32,20,32,36,24,42,42,28,48,57,
%T A113417 36,52,40,36,58,60,56,48,66,48,72,74,42,80,80,61,82,72,56,90,96,64,72,
%U A113417 98,70,100,104,64,106,108,72,114,96,84,144,111,84,104,128,84,130,144
%V A113417 1,-2,-4,8,7,-10,-12,8,18,-18,-16,24,21,-20,-28,32,20,-32,-36,24,42,-42,
               -28,48,57,-36,
%W A113417 -52,40,36,-58,-60,56,48,-66,-48,72,74,-42,-80,80,61,-82,-72,56,90,-96,
               -64,72,98,-70,
%X A113417 -100,104,64,-106,-108,72,114,-96,-84,144,111,-84,-104,128,84,-130,-144
%N A113417 Expansion of q^(-1)(eta(q^2)^2*eta(q^16))^2(eta(q^4)/eta(q^8))^3 in powers 
               of q^2.
%F A113417 Euler transform of period 8 sequence [ -2, -5, -2, -2, -2, -5, -2, -4, 
               ...].
%F A113417 a(n)=b(2n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = 
               (p^(e+1)-1)/(p-1) if p == 1, 7 (mod 8), b(p^e) = ((-p)^(e+1)-1)/(-p-1) 
               if p == 3, 5 (mod 8).
%F A113417 G.f.: Sum_{k>=0} a(k)x^(2k+1) = Sum_{k>0} (2k-1)*(-1)^[k/2]*x^(2k-1)/
               (1-x^(4k-2)) = x(Product_{k> 0} ((1-x^(2k))(1-x^(4k))(1+x^(8k)))^2/
               (1+x^(4k))).
%e A113417 q -2*q^3 -4*q^5 +8*q^7 +7*q^9 -10*q^11 -12*q^13 +8*q^15 +...
%o A113417 (PARI) a(n)=if(n<0, 0, sumdiv(2*n+1,d, d*(d%2)*(-1)^((d+1)\4)))
%o A113417 (PARI) {a(n)=local(A,p,e); if(n<0, 0, n=2*n+1; A=factor(n); prod(k=1,
               matsize(A)[1], if(p=A[k,1], e=A[k,2]; if(p==2, 0, p*=kronecker(2,
               p); (p^(e+1)-1)/(p-1)))))}
%o A113417 (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x+A)*eta(x^8+A))^2*(eta(x^2+A)/
               eta(x^4+A))^3, n))}
%Y A113417 Sequence in context: A065075 A001370 A039794 this_sequence A113419 A133007 
               A141532
%Y A113417 Adjacent sequences: A113414 A113415 A113416 this_sequence A113418 A113419 
               A113420
%K A113417 sign
%O A113417 0,2
%A A113417 Michael Somos, Oct 29 2005