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A261576 Expansion of 3 * b(q^2) * c(q^2) / c(q)^2 in powers of q where b(), c() are cubic AGM theta functions. 2
1, -2, -3, 12, -10, -18, 60, -48, -75, 228, -172, -252, 732, -524, -744, 2088, -1450, -1998, 5460, -3704, -4986, 13344, -8872, -11736, 30876, -20206, -26322, 68268, -44080, -56682, 145224, -92672, -117867, 298800, -188756, -237744, 597108, -373852, -466836 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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).

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 f(-q^2)^4 / (f(-q^3)^2 * f(q, q^2)^2) in powers of q where f(,) is Ramanujan's general theta function.

Expansion of (eta(q) * eta(q^2) * eta(q^6) / eta(q^3)^3)^2 in powers of q.

Euler transform of period 6 sequence [ -2, -4, 4, -4, -2, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 9/4 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A258099.

a(n) = A261326(2*n).  a(3*n + 2) = -3 * A233698(n).

EXAMPLE

G.f. = 1 - 2*x - 3*x^2 + 12*x^3 - 10*x^4 - 18*x^5 + 60*x^6 - 48*x^7 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ (QPochhammer[ q] QPochhammer[ q^2] QPochhammer[ q^6] / QPochhammer[ q^3]^3)^2, {q, 0, n}];

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^2 + A) * eta(x^6 + A) / eta(x^3 + A)^3)^2, n))};

CROSSREFS

Cf. A233698, A258099, A261326.

Sequence in context: A112979 A225723 A092972 * A081526 A075711 A079077

Adjacent sequences:  A261573 A261574 A261575 * A261577 A261578 A261579

KEYWORD

sign

AUTHOR

Michael Somos, Aug 25 2015

STATUS

approved

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Last modified December 5 16:12 EST 2019. Contains 329753 sequences. (Running on oeis4.)