A144558 Expansion of Product_{n >= 1} (1+q^(2*n-1))/((1-q^(4*n))*(1+q^(4*n-2))).

1, 1, -1, 0, 3, 2, -3, -1, 8, 5, -8, -3, 18, 11, -19, -7, 38, 22, -41, -16, 75, 42, -82, -33, 142, 78, -157, -64, 258, 138, -288, -120, 455, 239, -511, -215, 781, 404, -882, -374, 1310, 668, -1486, -635, 2153, 1084, -2450, -1053, 3477, 1733, -3967, -1712, 5524, 2726, -6316, -2737, 8652, 4233, -9907

Offset: 0

Author

N. J. A. Sloane, Jan 02 2009

Keywords

• sign

Comments

The authors of the article have informed me that there is a typo in the published g.f. - the factor (1+q^(4*n-2)) should be squared. When this is done, we get the sequence A085261. In short, this is an erroneous version of A085261.

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + x - x^2 + 3*x^4 + 2*x^5 - 3*x^6 - x^7 + 8*x^8 + 5*x^9 - 8*x^10 + ...
G.f. = 1/q + q^7 - q^15 + 3*q^31 + 2*q^39 - 3*q^47 - q^55 + 8*q^63 + 5*q^71 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• M. Ishikawa and J. Zeng, The Andrews-Stanley partition function and Al-Salam-Chihara polynomials, Disc. Math., 309 (2009), 151-175.
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Programs

Mathematica

QP = QPochhammer; s = QP[q^2]^3*(QP[q^8]/(QP[q]*QP[q^4]^4)) + O[q]^60; CoefficientList[s, q] (* Jean-François Alcover, Nov 14 2015, adapted from PARI *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^8 + A) / (eta(x + A) * eta(x^4 + A)^4), n))}; /* Michael Somos, Jun 04 2012 */

Formula

From Michael Somos, Jun 04 2012: (Start)

Expansion of chi(x) / f(x^2) in powers of x where chi(), f() are Ramanujan theta functions.

Expansion of q^(1/8) * eta(q^2)^3 * eta(q^8) / (eta(q) * eta(q^4)^4) in powers of q.

Euler transform of period 8 sequence [ 1, -2, 1, 2, 1, -2, 1, 1, ...]. (End)