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A259884 Expansion of phi(x) * f(-x^3)^3 / f(-x) in powers of x where phi(), f() are Ramanujan theta functions. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

M. 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: A268395 A304032 A022484 * A283972 A213509 A112180

Adjacent sequences:  A259881 A259882 A259883 * A259885 A259886 A259887

KEYWORD

nonn

AUTHOR

Michael Somos, Jul 07 2015

STATUS

approved

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Last modified February 19 16:02 EST 2019. Contains 320311 sequences. (Running on oeis4.)