A177398 O.g.f.: exp( Sum_{n>=1} (sigma(2n)-sigma(n))^2 * x^n/n ).

1, 4, 16, 64, 208, 656, 1984, 5632, 15520, 41476, 107312, 271232, 670464, 1622160, 3854208, 9003264, 20696640, 46895248, 104827472, 231353984, 504592448, 1088323584, 2322683072, 4908033280, 10273819136, 21313971876, 43843093488

Offset: 0

Author

Paul D. Hanna, May 30 2010

Keywords

• nonn

Comments

Here sigma(n) = A000203(n) is the sum of divisors of n. Compare g.f. to the formula for Jacobi theta_4(x) given by:

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

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

Examples

			G.f.: A(x) = 1 + 4*x + 16*x^2 + 64*x^3 + 208*x^4 + 656*x^5 +...
log(A(x)) = 4*x + 16*x^2/2 + 64*x^3/3 +...+ A054785(n)^2*x^n/n +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..400 from Paul D. Hanna)

Crossrefs

Cf. A054785, A000203, A177399.

Programs

Mathematica

nmax = 30; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, 2*k] - DivisorSigma[1,k])^2 * x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 26 2019 *)

PARI

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