A216406 G.f.: Product_{n>=1} ((1-x^n)/(1+x^n))^(2*n).

1, -4, 0, 8, 16, -8, -48, -56, 0, 116, 256, 264, -32, -648, -1296, -1392, -352, 2040, 5200, 7368, 6112, -784, -13744, -29304, -39648, -33804, -1376, 60368, 139552, 205304, 210208, 103432, -146528, -521744, -928480, -1190000, -1069904, -339720, 1110864, 3146640, 5278624

Offset

0,2

Comments

The number of contiguous signs seems to increase in proportion to the square-root of the number of terms.

Compare the g.f. to the Jacobi theta_4 series identity:

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

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..2500 from Paul D. Hanna)

Formula

G.f.: exp( Sum_{n>=1} -(sigma_2(2*n) - sigma_2(n))*x^n/n ) where sigma_2(n) = sum of squares of divisors of n.

Example

G.f.: A(x) = 1 - 4*x + 8*x^3 + 16*x^4 - 8*x^5 - 48*x^6 - 56*x^7 + 116*x^9 +...
where the g.f. equals the infinite product:
A(x) = (1-x)^2/(1+x)^2 * (1-x^2)^4/(1+x^2)^4 * (1-x^3)^6/(1+x^3)^6 * (1-x^4)^8/(1+x^4)^8 * (1-x^5)^10/(1+x^5)^10 *...
The logarithm of the g.f. is illustrated by:
-log(A(x)) = 4*x + 16*x^2/2 + 40*x^3/3 + 64*x^4/4 + 104*x^5/5 + 160*x^6/6 + 200*x^7/7 + 256*x^8/8 +...+ 4*A076577(n)*x^n/n +...

Prog

(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, -(sigma(2*m, 2)-sigma(m, 2))*x^m/m+x*O(x^n))), n)}
(PARI) {a(n)=polcoeff(prod(m=1, n, ((1-x^m)/(1+x^m +x*O(x^n)))^(2*m)), n)}
for(n=0, 100, print1(a(n), ", "))

Crossrefs

Cf. A156616, A076577, A001157 (sigma_2), A261386.

Sequence in context: A010769 A145831 A324482 * A152990 A177900 A214202

Adjacent sequences: A216403 A216404 A216405 * A216407 A216408 A216409

Keyword

sign

Author

Paul D. Hanna, Sep 06 2012

Status

approved