A195584 O.g.f.: exp( Sum_{n>=1} (sigma(2*n^2)-sigma(n^2)) * x^n/n ).

1, 2, 6, 18, 42, 102, 238, 522, 1130, 2394, 4926, 9978, 19890, 38942, 75254, 143598, 270506, 504126, 929926, 1698322, 3074010, 5516898, 9820550, 17349554, 30430610, 53007162, 91734262, 157771538, 269734714, 458542822, 775281982, 1303971722, 2182227546, 3634444634

Offset: 0

Author

Paul D. Hanna, Sep 20 2011

Keywords

• nonn

Comments

Compare g.f. to the formula for Jacobi theta_4(x) given by:

theta_4(x) = exp( Sum{n>=1} -(sigma(2*n)-sigma(n))*x^n/n )

where theta_4(x) = 1 + Sum_{n>=1} 2*(-x)^(n^2).

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

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 18*x^3 + 42*x^4 + 102*x^5 + 238*x^6 +...
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 +...+ A195585(n)*x^n/n +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Crossrefs

Cf. A195585, A215603, A177399, A015128, A054785; variant: A225958.

Programs

Mathematica

nmax = 40; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, 2*n^2] - DivisorSigma[1, n^2])*(x^n/n), {n, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2015 *)

PARI

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

Formula

O.g.f.: exp( Sum_{n>=1} A054785(n^2)*x^n/n ), where exp( Sum_{n>=1} A054785(n)*x^n/n ) = 1/(1+2*Sum_{n>=1} (-x)^(n^2)), which is the g.f. of A015128.