A213592 Expansion of q^(-1/3) * phi(q^2) * c(q) / 3 in powers of q where phi() is a Ramanujan theta function and c() is a cubic AGM theta function.

1, 1, 4, 2, 6, 1, 6, 2, 7, 4, 8, 4, 10, 2, 10, 0, 9, 6, 12, 6, 10, 1, 14, 4, 16, 6, 8, 8, 12, 2, 12, 0, 20, 7, 20, 6, 10, 4, 20, 6, 11, 8, 16, 8, 20, 4, 14, 0, 20, 10, 12, 8, 26, 2, 22, 6, 15, 10, 20, 12, 18, 0, 28, 0, 20, 9, 20, 14, 16, 6, 10, 6, 24, 12, 32

Offset

0,3

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^(-1/3) * eta(q^3)^3 * eta(q^4)^5 / (eta(q) * eta(q^2)^2 * eta(q^8)^2) in powers of q.

Euler transform of period 24 sequence [ 1, 3, -2, -2, 1, 0, 1, 0, -2, 3, 1, -5, 1, 3, -2, 0, 1, 0, 1, -2, -2, 3, 1, -3, ...].

a(16*n + 15) = 0. a(4*n + 1) = a(n).

Example

1 + x + 4*x^2 + 2*x^3 + 6*x^4 + x^5 + 6*x^6 + 2*x^7 + 7*x^8 + 4*x^9 + ...
q + q^4 + 4*q^7 + 2*q^10 + 6*q^13 + q^16 + 6*q^19 + 2*q^22 + 7*q^25 + ...

Mathematica

 QP := QPochhammer; a[n_]:= SeriesCoefficient[(QP[q^3]^3*QP[q^4]^5)/( QP[q]*QP[q^2]^2*QP[q^8]^2), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Jan 07 2018 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^3 * eta(x^4 + A)^5 / (eta(x + A) * eta(x^2 + A)^2 * eta(x^8 + A)^2), n))}

Crossrefs

Sequence in context: A131749 A016515 A011306 * A019719 A205111 A340177

Adjacent sequences: A213589 A213590 A213591 * A213593 A213594 A213595

Keyword

nonn

Author

Michael Somos, Jun 15 2012

Status

approved