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

0,4

Comments

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

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

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).

Example

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 + ...

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 *)

Prog

(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))}

Crossrefs

Cf. A008443.

Sequence in context: A284051 A226743 A166269 * A182910 A055460 A067514

Adjacent sequences: A181645 A181646 A181647 * A181649 A181650 A181651

Keyword

nonn

Author

Michael Somos, Jun 22 2012

Status

approved