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A161800 G.f.: A(q) = exp( Sum_{n>=1} A002129(n) * 2*A006519(n) * q^n/n ).

%I A161800 #2 Mar 30 2012 18:37:17
%S A161800 1,2,0,0,-6,-16,0,0,-8,18,0,0,112,176,0,0,-86,-544,0,0,-752,-160,0,0,
%T A161800 1360,2834,0,0,1216,-5104,0,0,-5384,3232,0,0,10762,18032,0,0,-8176,
%U A161800 -68992,0,0,-59888,48400,0,0,130160,143074,0,0,47696,-343088,0,0
%N A161800 G.f.: A(q) = exp( Sum_{n>=1} A002129(n) * 2*A006519(n) * q^n/n ).
%C A161800 A002129 forms the l.g.f. of log[ Sum_{n>=0} q^(n(n+1)/2) ], while
%C A161800 2*A006519 forms the l.g.f. of binary partitions (A000123) and
%C A161800 A006519(n) is the highest power of 2 dividing n.
%F A161800 a(n) = 0 when n == 2 or 3 (mod 4).
%F A161800 Define the nonzero series QUADRASECTIONS:
%F A161800 Q_0(q) = Sum_{n>=0} a(4n)*q^n,
%F A161800 Q_1(q) = Sum_{n>=0} a(4n+1)*q^n, then:
%F A161800 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.
%F A161800 [The above statements are conjectures needing proof.]
%e A161800 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 +...
%e A161800 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 +...
%e A161800 Sum_{n>=1} A002129(n)*q^n/n = log(1 + q + q^3 + q^6 + q^10 + q^15 +...),
%e A161800 Sum_{n>=1} 2*A006519(n)*x^n/n = log of the g.f. of binary partitions A000123.
%e A161800 QUADRASECTIONS:
%e A161800 Q_0(q) = 1 - 6*q - 8*q^2 + 112*q^3 - 86*q^4 - 752*q^5 + 1360*q^6 +...
%e A161800 Q_1(q) = 2 - 16*q + 18*q^2 + 176*q^3 - 544*q^4 - 160*q^5 + 2834*q^6 +...
%e A161800 The ratio Q_1(q)/Q_0(q) yields:
%e A161800 2 - 4*q + 10*q^2 - 20*q^3 + 36*q^4 - 64*q^5 + 110*q^6 - 180*q^7 +...
%e A161800 which appears to equal the g.f. of A127392.
%o A161800 (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)}
%Y A161800 Cf. A127392, quadrasections: A161801, A161802.
%Y A161800 Cf. A002129, A006519, A000123.
%K A161800 sign
%O A161800 0,2
%A A161800 _Paul D. Hanna_, Jul 19 2009