A181648 Expansion of x^(-2/3) * psi(x) * c(x^2) / 3 in powers of x where psi() is a Ramanujan theta function and c() is a cubic AGM theta function.

1, 1, 1, 2, 2, 3, 1, 2, 3, 2, 4, 3, 3, 3, 4, 3, 2, 2, 6, 5, 3, 5, 3, 5, 4, 5, 3, 4, 5, 4, 5, 4, 5, 7, 6, 7, 3, 3, 7, 4, 8, 4, 4, 5, 7, 6, 5, 6, 7, 8, 6, 4, 6, 9, 6, 8, 6, 4, 4, 4, 11, 7, 4, 11, 4, 9, 6, 7, 8, 7, 11, 5, 5, 8, 8, 10, 6, 5, 10, 6, 8, 6, 7, 7, 8

Offset: 0

Author

Michael Somos, Jun 22 2012

Keywords

• nonn

Comments

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

Examples

			1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + x^6 + 2*x^7 + 3*x^8 + 2*x^9 + 4*x^10 + ...
q^19 + q^43 + q^67 + 2*q^91 + 2*q^115 + 3*q^139 + q^163 + 2*q^187 + 3*q^211 + ...
		

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. A008443.

Programs

Mathematica

A181648[n_]:= SeriesCoefficient[QPochhammer[q^2, q^2]*QPochhammer[q^6, q^6]^3/QPochhammer[q, q], {q, 0, n}]; Table[A181648[n], {n, 0, 50}] (* G. C. Greubel, Dec 24 2017 *)

PARI

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

Formula

Expansion of q^(-19/24) * eta(q^2) * eta(q^6)^3 / eta(q) in powers of q.

Euler transform of period 6 sequence [ 1, 0, 1, 0, 1, -3, ...].

3 * a(n) = A008443(3*n + 2).