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A227498 Expansion of (1/q) * (f(q) / f(q^9))^3 in powers of q where f() is a Ramanujan theta function. 2
1, 3, 0, -5, 0, 0, -7, 0, 0, -3, 0, 0, 15, 0, 0, 32, 0, 0, 9, 0, 0, -58, 0, 0, -96, 0, 0, -22, 0, 0, 149, 0, 0, 253, 0, 0, 68, 0, 0, -372, 0, 0, -599, 0, 0, -140, 0, 0, 826, 0, 0, 1317, 0, 0, 317, 0, 0, -1768, 0, 0, -2735, 0, 0, -632, 0, 0, 3526, 0, 0, 5434 (list; graph; refs; listen; history; text; internal format)
OFFSET

-1,2

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 = -1..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of -3 * b(-q) / c(-q^3) in powers of q where b(), c() are cubic AGM theta functions.

Expansion of ( eta(q^2)^3 * eta(q^9) * eta(q^36) / (eta(q) * eta(q^4) * eta(q^18)^3) )^3 in powers of q.

Euler transform of period 36 sequence [3, -6, 3, -3, 3, -6, 3, -3, 0, -6, 3, -3, 3, -6, 3, -3, 3, 0, 3, -3, 3, -6, 3, -3, 3, -6, 0, -3, 3, -6, 3, -3, 3, -6, 3, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 27 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A227454.

G.f.: (1/x) * (Product_{k>0} (1 - (-x)^k) / (1 - (-x)^(9*k)))^3.

a(3*n) = 0 unless n=0. a(3*n + 1) = 0. a(3*n-1) = (-1)^n * A058091(n).

Convolution inverse of A227454.

EXAMPLE

G.f. = 1/q + 3 - 5*q^2 - 7*q^5 - 3*q^8 + 15*q^11 + 32*q^14 + 9*q^17 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ (QPochhammer[ -q] / QPochhammer[ -q^9])^3 / q, {q, 0, n}];

PROG

(PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( ( eta(x^2 + A)^3 * eta(x^9 + A) * eta(x^36 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^18 + A)^3))^3, n))};

CROSSREFS

Cf. A058091, A227454.

Sequence in context: A133089 A198954 A136599 * A131986 A002656 A234434

Adjacent sequences:  A227495 A227496 A227497 * A227499 A227500 A227501

KEYWORD

sign

AUTHOR

Michael Somos, Sep 22 2013

STATUS

approved

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Last modified April 22 19:11 EDT 2021. Contains 343177 sequences. (Running on oeis4.)