%I A113419
%S A113419 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 A113419 36,52,40,36,58,60,56,48,66,48,72,74,42,80,80,61,82,72,56,90,96,64,72,
%U A113419 98,70,100,104,64,106,108,72,114,96,84,144,111,84,104,128,84,130,144
%V A113419 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 A113419 -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 A113419 -100,-104,64,106,-108,-72,114,96,-84,-144,111,84,-104,-128,84,130,-144
%N A113419 Expansion of q^(-1)(eta(q^4)^9*eta(q^16)^2)/(eta(q^2)^2*eta(q^8)^5) in 
               powers of q^2.
%F A113419 Euler transform of period 8 sequence [2, -7, 2, -2, 2, -7, 2, -4, ...].
%F A113419 a(n)=b(2n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = 
               (x^(e+1)-y^(e+1))/(x-y) where x=p*(-1)^[p/4] and y = (-1)^[p/2].
%F A113419 G.f.: Sum_{k>0} (2k-1)*(-1)^[(k-1)/2]*x^(2k-1)/(1+x^(4k-2)).
%e A113419 q +2*q^3 -4*q^5 -8*q^7 +7*q^9 +10*q^11 -12*q^13 -8*q^15 +...
%o A113419 (PARI) a(n)=if(n<0, 0, n=2*n+1; sumdiv(n,d, d*(d%2)*(-1)^((n/d)\2+(d-1)\4)))
%o A113419 (PARI) {a(n)=local(A,p,e,t); 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, t=(-1)^(p\2); p*=kronecker(-2,
               p); (p^(e+1)-t^(e+1))/(p-t)))))}
%o A113419 (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^9*eta(x^8+A)^2/
               (eta(x+A)^2*eta(x^4+A)^5), n))}
%Y A113419 A113417(n) = (-1)^n*a(n).
%Y A113419 Sequence in context: A001370 A039794 A113417 this_sequence A133007 A141532 
               A083550
%Y A113419 Adjacent sequences: A113416 A113417 A113418 this_sequence A113420 A113421 
               A113422
%K A113419 sign
%O A113419 0,2
%A A113419 Michael Somos, Oct 29 2005