%I A115607
%S A115607 1,1,4,1,6,4,8,1,13,6,12,4,14,8,24,1,18,13,20,6,32,12,24,4,31,14,40,8,
%T A115607 30,24,32,1,48,18,48,13,38,20,56,6,42,32,44,12,78,24,48,4,57,31,72,14,
%U A115607 54,40,72,8,80,30,60,24,62,32,104,1,84,48,68,18,96,48,72,13,74,38,124
%V A115607 1,-1,4,-1,6,-4,8,-1,13,-6,12,-4,14,-8,24,-1,18,-13,20,-6,32,-12,24,-4,
               31,-14,40,-8,30,
%W A115607 -24,32,-1,48,-18,48,-13,38,-20,56,-6,42,-32,44,-12,78,-24,48,-4,57,-31,
               72,-14,54,-40,
%X A115607 72,-8,80,-30,60,-24,62,-32,104,-1,84,-48,68,-18,96,-48,72,-13,74,-38,
               124
%N A115607 Sum of odd divisors of n times (-1)^(n+1).
%D A115607 C. G. J. Jacobi, Gesammelte Werke, vol. 2, 1969, Chelsea, NY, p. 179 
               eq (6.)
%F A115607 Moebius transform is [1, -2, 3, 0, 5, -6, 7, 0, 9, -10, 11, 0, ...].
%F A115607 a(n) is multiplicative with a(2^e) = -1 if e>0, a(p^e) = (p(e+1)-1)/(p-1) 
               if p>2.
%F A115607 G.f.: (theta_2(q)^4 - theta_4(q)^4 +1)/24.
%F A115607 G.f.: Sum_{k>0} -k(-x)^k/(1+(-x)^k) = Sum_{k>0} x^k/(1-(-x)^k)^2.
%F A115607 G.f.: Sum_{k>0} (2k-1)*x^(2k-1)/(1+x^(2k-1)) = Sum_{k>0} x^k/(1-x^k)^2 
               - 4*x^(4k-2)/(1-x^(4k-2))^2.
%o A115607 (PARI) a(n)=if(n<1, 0, sumdiv(n,d, (-1)^(n+d)*n/d))
%o A115607 (PARI) a(n)=if(n<1, 0, -(-1)^n*sumdiv(n,d,d%2*d))
%Y A115607 Cf. a(n)=-(-1)^n*A000593(n). A103640(n)=-24*a(n) if n>0.
%Y A115607 Sequence in context: A117001 A098986 A000593 this_sequence A076717 A120422 
               A110312
%Y A115607 Adjacent sequences: A115604 A115605 A115606 this_sequence A115608 A115609 
               A115610
%K A115607 sign,mult
%O A115607 1,3
%A A115607 Michael Somos, Jan 26 2006