A215603 O.g.f.: exp( Sum_{n>=1} -(sigma(2*n^2) - sigma(n^2)) * (-x)^n/n ).

1, 2, -2, 2, 10, -10, 6, 10, -22, 58, -58, 10, 114, -210, 270, -242, 74, 382, -930, 1474, -1542, 1010, 446, -2798, 5682, -7718, 8030, -5182, -998, 11126, -23802, 35626, -42246, 39450, -20810, -15546, 69514, -133770, 194918, -234106, 227410, -147706, -19738, 282234

Offset: 0

Author

Paul D. Hanna, Aug 17 2012

Keywords

• sign

Comments

Compare to the Jacobi theta_3 function:

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

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

Examples

			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 +...
where
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 +...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1000

Crossrefs

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

Programs

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)}
for(n=0,50,print1(a(n),", "))

Formula

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