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A261240 Expansion of f(-x^6, -x^12)^2 / (f(-x, -x) * f(-x^3, -x^15)) in powers of x where f(, ) is Ramanujan's general theta function. 3
1, 2, 4, 9, 16, 28, 47, 76, 120, 185, 280, 416, 608, 878, 1252, 1765, 2464, 3408, 4676, 6364, 8600, 11545, 15400, 20424, 26938, 35346, 46152, 59981, 77616, 100016, 128369, 164140, 209120, 265510, 335992, 423840, 533035, 668404, 835804, 1042308, 1296448 (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).

LINKS

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

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015.

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of f(-x^6) * psi(x^3) / (phi(-x) * psi(x^9)) in powers of x where phi(), psi(), f() are Ramanujan theta functions.

Expansion of q^(1/2) * eta(q^2) * eta(q^6)^3 * eta(q^9) / (eta(q)^2 * eta(q^3) * eta(q^18)^2) in powers of q.

Euler transform of period 18 sequence [2, 1, 3, 1, 2, -1, 2, 1, 2, 1, 2, -1, 2, 1, 3, 1, 2, 0, ...].

a(n) = A058647(2*n - 1) = A186115(2*n - 1) = A186964(2*n - 1) = A187020(2*n - 1).

a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(7/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015

EXAMPLE

G.f. = 1 + 2*x + 4*x^2 + 9*x^3 + 16*x^4 + 28*x^5 + 47*x^6 + 76*x^7 + ...

G.f. = q^-1 + 2*q + 4*q^3 + 9*q^5 + 16*q^7 + 28*q^9 + 47*q^11 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ x^(3/4) QPochhammer[ x^6] EllipticTheta[ 2, 0, x^(3/2)] / (EllipticTheta[ 4, 0, x] EllipticTheta[ 2, 0, x^(9/2)]), {x, 0, n}];

nmax=60; CoefficientList[Series[Product[(1+x^k) * (1-x^(6*k))^3 * (1-x^(9*k)) / ((1-x^k) * (1-x^(3*k)) * (1-x^(18*k))^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 13 2015 *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^6 + A)^3 * eta(x^9 + A) / (eta(x + A)^2 * eta(x^3 + A) * eta(x^18 + A)^2), n))};

CROSSREFS

Cf. A058647, A186115, A186964, A187020.

Sequence in context: A023194 A114080 A090676 * A000291 A081055 A034446

Adjacent sequences:  A261237 A261238 A261239 * A261241 A261242 A261243

KEYWORD

nonn

AUTHOR

Michael Somos, Aug 12 2015

STATUS

approved

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Last modified October 18 22:14 EDT 2019. Contains 328211 sequences. (Running on oeis4.)