A215603 - OEIS

%I #10 May 22 2013 00:14:10

%S 1,2,-2,2,10,-10,6,10,-22,58,-58,10,114,-210,270,-242,74,382,-930,

%T 1474,-1542,1010,446,-2798,5682,-7718,8030,-5182,-998,11126,-23802,

%U 35626,-42246,39450,-20810,-15546,69514,-133770,194918,-234106,227410,-147706,-19738,282234

%N O.g.f.: exp( Sum_{n>=1} -(sigma(2*n^2) - sigma(n^2)) * (-x)^n/n ).

%C Compare to the Jacobi theta_3 function:

%C 1 + 2*Sum_{n>=1} x^(n^2)  =  exp( Sum_{n>=1} -(sigma(2*n) - sigma(n))*(-x)^n/n ).

%C Here sigma(n) = A000203(n) is the sum of divisors of n.

%H Paul D. Hanna, <a href="/A215603/b215603.txt">Table of n, a(n) for n = 0..1000</a>

%F O.g.f.: exp( Sum_{n>=1} -A054785(n^2)*(-x)^n/n ), where A054785(n^2) = A195585(n).

%e O.g.f.: A(x) = 1 + 2*x - 2*x^2 + 2*x^3 + 10*x^4 - 10*x^5 + 6*x^6 + 10*x^7 +...

%e where

%e log(A(x)) = 2*x - 8*x^2/2 + 26*x^3/3 - 32*x^4/4 + 62*x^5/5 - 104*x^6/6 + 114*x^7/7 - 128*x^8/8 + 242*x^9/9 - 248*x^10/10 + 266*x^11/11 - 416*x^12/12 +...+ -A054785(n^2)*(-x)^n/n +...

%o (PARI) {a(n)=polcoeff(exp(sum(m=1, n,-(sigma(2*m^2)-sigma(m^2))*(-x)^m/m)+x^2*O(x^n)),n)}

%o for(n=0,50,print1(a(n),", "))

%Y Cf. A195584, A195585, A177399, A054785, A000203; variants: A225925, A225957.

%K sign

%O 0,2

%A _Paul D. Hanna_, Aug 17 2012