%I A111938
%S A111938 1,2,0,4,10,0,0,8,9,20,0,0,26,0,0,16,34,18,0,40,0,0,0,0,75,52,0,0,58,0,
%T A111938 0,32,0,68,0,36,74,0,0,80,82,0,0,0,90,0,0,0,49,150,0,104,106,0,0,0,0,
%U A111938 116,0,0,122,0,0,64,260,0,0,136,0,0,0,72,146,148,0,0,0,0,0,160,81,164
%N A111938 a(n) = n times number of divisors of n of form 4m+1 - n times number 
               of divisors of form 4m+3.
%F A111938 Multiplicative with a(p^e) = p^e if p = 2; (e+1)p^e if p == 1 (mod 4); 
               (1+(-1)^e)/2 p^e if p == 3 (mod 4).
%F A111938 G.f.: Sum_{k>0} k(x^k-x^(3k))/(1+x^(2k))^2 = Sum_{k>0} -(-1)^k(2k-1)x^(2k-1)/
               (1-x^(2k-1))^2.
%F A111938 G.f.: xd/dx(theta_3(x)^2)/4 . - Michael Somos Nov 07 2005
%F A111938 G.f.: (1/4)* Sum_{u,v} (u*u +v*v)* x^(u*u +v*v). - Michael Somos Jun 
               14 2007
%o A111938 (PARI) a(n)=if(n<1, 0, n*sumdiv(n,d, (d%4==1)-(d%4==3)))
%o A111938 (PARI) {a(n)=local(r); if(n<1, 0, r=sqrtint(n); sum(x=-r,r, sum(y=-r,
               r, if(x^2+y^2==n, (x+y)^2) ))/4 )} /* Michael Somos Sep 12 2005 */
%o A111938 (PARI) {a(n)=if(n<1, 0, n*polcoeff( sum(k=1,sqrtint(n), 2*x^k^2, 1+x*O(x^n))^2, 
               n)/4 )} /* Michael Somos Sep 12 2005 */
%Y A111938 n*A002654(n)=a(n).
%Y A111938 Sequence in context: A021492 A077119 A002938 this_sequence A167341 A055978 
               A069025
%Y A111938 Adjacent sequences: A111935 A111936 A111937 this_sequence A111939 A111940 
               A111941
%K A111938 nonn,mult
%O A111938 1,2
%A A111938 Michael Somos, Aug 21 2005