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

1, 2, 0, 0, -6, -16, 0, 0, -8, 18, 0, 0, 112, 176, 0, 0, -86, -544, 0, 0, -752, -160, 0, 0, 1360, 2834, 0, 0, 1216, -5104, 0, 0, -5384, 3232, 0, 0, 10762, 18032, 0, 0, -8176, -68992, 0, 0, -59888, 48400, 0, 0, 130160, 143074, 0, 0, 47696, -343088, 0, 0

Offset

0,2

Comments

A002129 forms the l.g.f. of log[ Sum_{n>=0} q^(n(n+1)/2) ], while

2*A006519 forms the l.g.f. of binary partitions (A000123) and

A006519(n) is the highest power of 2 dividing n.

Links

Table of n, a(n) for n=0..55.

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 +...),
Sum_{n>=1} 2*A006519(n)*x^n/n = log of the g.f. of binary partitions A000123.
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

Cf. A127392, quadrasections: A161801, A161802.

Cf. A002129, A006519, A000123.

Sequence in context: A244133 A348639 A244142 * A246608 A100344 A370796

Adjacent sequences: A161797 A161798 A161799 * A161801 A161802 A161803

Keyword

sign

Author

Paul D. Hanna, Jul 19 2009

Status

approved