A259884 Expansion of phi(x) * f(-x^3)^3 / f(-x) in powers of x where phi(), f() are Ramanujan theta functions.

1, 3, 4, 4, 4, 7, 8, 4, 5, 8, 12, 8, 4, 12, 8, 8, 9, 12, 16, 4, 12, 15, 8, 8, 8, 20, 20, 8, 8, 12, 16, 16, 8, 15, 20, 12, 12, 16, 16, 12, 13, 24, 20, 8, 8, 24, 24, 8, 16, 12, 28, 16, 12, 28, 8, 20, 13, 20, 28, 16, 20, 24, 16, 8, 8, 27, 36, 12, 16, 28, 24, 12

Offset

0,2

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 phi(x) * c(x) * (1/3) * x^(-1/3) in powers of x where phi() is a Ramanujan theta function (A000122) and c() is a cubic AGM theta function (A005882).

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

Euler transform of period 12 sequence [3, -2, 0, 0, 3, -5, 3, 0, 0, -2, 3, -3, ...].

Example

G.f. = 1 + 3*x + 4*x^2 + 4*x^3 + 4*x^4 + 7*x^5 + 8*x^6 + 4*x^7 + 5*x^8 + ...
G.f. = q + 3*q^4 + 4*q^7 + 4*q^10 + 4*q^13 + 7*q^16 + 8*q^19 + 4*q^22 + ...

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_] := SeriesCoefficient[q^(-1/3)* eta[q^2]^5*eta[q^3]^3/(eta[q]^3*eta[q^4]^2), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Mar 16 2018 *)
a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, q]*QPochhammer[q^3]^3 /QPochhammer[q], {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Mar 18 2018 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^3 / (eta(x + A)^3 * eta(x^4 + A)^2), n))};
(PARI) q='q+O('q^99); Vec(eta(q^2)^5*eta(q^3)^3/(eta(q)^3*eta(q^4)^2)) \\ Altug Alkan, Mar 18 2018

Crossrefs

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

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

Sequence in context: A361497 A352717 A359807 * A283972 A213509 A112180

Adjacent sequences: A259881 A259882 A259883 * A259885 A259886 A259887

Keyword

nonn

Author

Michael Somos, Jul 07 2015

Status

approved