OFFSET
0,2
COMMENTS
FORMULA
a(n) = 0 when n == 2 or 3 (mod 4).
Define the nonzero series QUADRASECTIONS:
Q_0(q) = Sum_{n>=0} a(4n)*q^n,
Q_1(q) = Sum_{n>=0} a(4n+1)*q^n, then:
Q_1(q)/Q_0(q) = series expansion of the elliptic function sqrt(k)/q^(1/4), where sqrt(k) = theta_2/theta_3, as described by A127392.
[The above statements are conjectures needing proof.]
EXAMPLE
G.f.: A(q) = 1 + 2*q - 6*q^4 - 16*q^5 - 8*q^8 + 18*q^9 + 112*q^12 + 176*q^13 +...
log(A(q)) = 2*q - 4*q^2/2 + 8*q^3/3 - 40*q^4/4 + 12*q^5/5 - 16*q^6/6 +...
Sum_{n>=1} A002129(n)*q^n/n = log(1 + q + q^3 + q^6 + q^10 + q^15 +...),
QUADRASECTIONS:
Q_0(q) = 1 - 6*q - 8*q^2 + 112*q^3 - 86*q^4 - 752*q^5 + 1360*q^6 +...
Q_1(q) = 2 - 16*q + 18*q^2 + 176*q^3 - 544*q^4 - 160*q^5 + 2834*q^6 +...
The ratio Q_1(q)/Q_0(q) yields:
2 - 4*q + 10*q^2 - 20*q^3 + 36*q^4 - 64*q^5 + 110*q^6 - 180*q^7 +...
which appears to equal the g.f. of A127392.
PROG
(PARI) {a(n)=local(L=sum(m=1, n, 2*2^valuation(m, 2)*sumdiv(m, d, -(-1)^d*d)*x^m/m)+x*O(x^n)); polcoeff(exp(L), n)}
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jul 19 2009
STATUS
approved