A227732 O.g.f.: exp( Sum_{n>=1} (sigma(2*n)^2 - sigma(n)^2) * x^n/n ).

1, 8, 52, 288, 1396, 6208, 25744, 100608, 374500, 1336488, 4596000, 15297056, 49444368, 155640640, 478268800, 1437600000, 4234216836, 12238666208, 34761065924, 97130259232, 267280386128, 724987680384, 1940011007056, 5125212451584, 13376644454672, 34512562565224

Offset

0,2

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.

Links

Table of n, a(n) for n=0..25.

Formula

Logarithmic derivative yields A227733.

Example

G.f.: A(x) = 1 + 8*x + 52*x^2 + 288*x^3 + 1396*x^4 + 6208*x^5 + 25744*x^6 +...
where
log(A(x)) = 8*x + 40*x^2/2 + 128*x^3/3 + 176*x^4/4 + 288*x^5/5 + 640*x^6/6 +...+ A227733(n)*x^n/n +...

Prog

(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, 30, print1(a(n), ", "))

Crossrefs

Cf. A227733.

Sequence in context: A193427 A022732 A256047 * A000432 A153336 A080279

Adjacent sequences: A227729 A227730 A227731 * A227733 A227734 A227735

Keyword

nonn

Author

Paul D. Hanna, Jul 24 2013

Status

approved