A208457 Expansion of x * f(-x) * f(-x^12)^3 * psi(-x^3) / psi(x^2) in powers of x where psi(), f() are Ramanujan theta functions.

0, 1, -1, -2, 0, 3, 2, -4, 0, 5, -1, -8, 0, 7, 4, -8, 0, 9, -8, -10, 0, 14, 6, -12, 0, 16, -6, -14, 0, 15, 8, -20, 0, 17, -14, -18, 0, 19, 10, -24, 0, 26, -1, -22, 0, 23, 16, -28, 0, 25, -20, -32, 0, 32, 14, -28, 0, 29, -12, -30, 0, 38, 16, -32, 0, 33, -31

Offset: 0

Author

Michael Somos, Feb 26 2012

Keywords

• sign

Comments

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

Examples

			G.f. = x - x^2 - 2*x^3 + 3*x^5 + 2*x^6 - 4*x^7 + 5*x^9 - x^10 - 8*x^11 + ...
G.f. = q^5 - q^8 - 2*q^11 + 3*q^17 + 2*q^20 - 4*q^23 + 5*q^29 - q^32 - 8*q^35 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A208435.

Programs

Mathematica

QP:= QPochhammer; Join[{0}, CoefficientList[Series[Simplify[QP[q]* QP[q^2]*QP[q^3]*QP[q^12]^4/(QP[q^4]^2*QP[q^6]), q > 0], {q, 0, 50}], q]] (* G. C. Greubel, Aug 12 2018 *)

PARI

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)^4 / (eta(x^4 + A)^2 * eta(x^6 + A)), n))};

Formula

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

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

a(4*n) = 0. a(n) = -(-1)^n * A208435(n).