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A259827 Expansion of phi(x) * f(-x^12)^3 / f(-x^4) in powers of x where phi(), f() are Ramanujan theta functions. 2
1, 2, 0, 0, 3, 2, 0, 0, 4, 6, 0, 0, 4, 2, 0, 0, 4, 8, 0, 0, 7, 2, 0, 0, 8, 10, 0, 0, 4, 4, 0, 0, 5, 10, 0, 0, 8, 4, 0, 0, 12, 10, 0, 0, 8, 6, 0, 0, 4, 14, 0, 0, 12, 2, 0, 0, 8, 14, 0, 0, 8, 4, 0, 0, 9, 18, 0, 0, 12, 6, 0, 0, 16, 14, 0, 0, 4, 4, 0, 0, 12, 12 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

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

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..2500

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of phi(x) * c(x^4) / (3 * x^(4/3)) in powers of x where phi() is a Ramanujan theta function and c() is a cubic AGM theta function.

Expansion of q^(-4/3) * eta(q^2)^5 * eta(q^12)^3 / (eta(q)^2 * eta(q^4)^3) in powers of q.

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

a(4*n + 2) = a(4*n + 3) = 0. a(4*n + 1) = 2 * A259655(n). 6 * a(n) = A259825(3*n + 4).

EXAMPLE

G.f. = 1 + 2*x + 3*x^4 + 2*x^5 + 4*x^8 + 6*x^9 + 4*x^12 + 2*x^13 + ...

G.f. = q^4 + 2*q^7 + 3*q^16 + 2*q^19 + 4*q^28 + 6*q^31 + 4*q^40 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ x^12]^3 / QPochhammer[ x^4], {x, 0, n}];

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^12 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)^3), n))};

CROSSREFS

Cf. A259655, A259825.

Sequence in context: A261115 A216229 A224777 * A143161 A225853 A142886

Adjacent sequences:  A259824 A259825 A259826 * A259828 A259829 A259830

KEYWORD

nonn

AUTHOR

Michael Somos, Jul 05 2015

STATUS

approved

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Last modified December 10 04:15 EST 2019. Contains 329885 sequences. (Running on oeis4.)