A230256 Expansion of f(-x) * psi(x^2) * phi(x^3) / f(-x^3)^3 in powers of x where phi(), psi(), f() are Ramanujan theta functions.

1, -1, 0, 4, -6, 1, 11, -19, 4, 31, -50, 11, 77, -122, 28, 173, -273, 62, 370, -573, 130, 751, -1149, 261, 1461, -2214, 498, 2750, -4125, 923, 5022, -7472, 1663, 8936, -13202, 2919, 15551, -22817, 5019, 26521, -38681, 8467, 44417, -64438, 14035, 73197

Offset: 0

Author

Michael Somos, Oct 14 2013

Keywords

• sign

Comments

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

Examples

			G.f. = 1 - x + 4*x^3 - 6*x^4 + x^5 + 11*x^6 - 19*x^7 + 4*x^8 + 31*x^9 + ...
G.f. = q^-1 - q^11 + 4*q^35 - 6*q^47 + q^59 + 11*q^71 - 19*q^83 + 4*q^95 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A062242, A062244, A132179, A132301, A182032, A182036, A182056.

Cf. A058531, A093073, A132976, A143840, A164268, A164612, A182033, A193261, A254372.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] EllipticTheta[ 3, 0, x^3] QPochhammer[ x] / (2 x^(1/4) QPochhammer[ x^3]^3), {x, 0, n}]; (* Michael Somos, Jan 29 2015 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A)^2 * eta(x^6 + A)^5 / (eta(x^2 + A) * eta(x^3 + A)^5 * eta(x^12 + A)^2), n))};

Formula

Expansion of q^(1/12) * eta(q) * eta(q^4)^2 * eta(q^6)^5 / (eta(q^2) * eta(q^3)^5 * eta(q^12)^2) in powers of q.

Euler transform of period 12 sequence [ -1, 0, 4, -2, -1, 0, -1, -2, 4, 0, -1, 0, ...].

a(n) = A132179(2*n) = A062242(4*n) = A062244(4*n) = A132301(4*n) = A182056(4*n) = A182036(6*n) = A182032(12*n - 1).

a(n) = A058531(12*n) = A093073(12*n) = A132976(12*n) = A143840(12*n) = A164268(12*n) = A164612(12*n) = A182033(12*n) = A193261(12*n). - Michael Somos, Jan 29 2015