A225958 O.g.f.: exp( Sum_{n>=1} (sigma(2*n^3) - sigma(n^3)) * x^n/n ).

1, 2, 10, 44, 134, 468, 1524, 4584, 13862, 40566, 114880, 321052, 879092, 2360156, 6248864, 16297384, 41902454, 106437600, 267149022, 662979572, 1628437160, 3960377672, 9541519732, 22786066280, 53958062564, 126750346970, 295476011176, 683776368416, 1571299804688

Offset: 0

Author

Paul D. Hanna, May 21 2013

Keywords

• nonn

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), the sum of the divisors of n.

Examples

			O.g.f.: A(x) = 1 + 2*x + 10*x^2 + 44*x^3 + 134*x^4 + 468*x^5 + 1524*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 + 114*x^7/7 + 128*x^8/8 + 242*x^9/9 + 248*x^10/10 + 266*x^11/11 +...+ A054785(n^3)*x^n/n +...
		

Crossrefs

Cf. A225957, A225959, A054785, A000203; variants: A195584, A215603, A225925, A224902.

Programs

PARI

{a(n)=polcoeff(exp(sum(m=1, n, (sigma(2*m^3)-sigma(m^3))*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^3)*x^n/n ).

Logarithmic derivative equals A225959.