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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.
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%I #16 Mar 12 2021 22:24:48

%S 1,2,4,9,16,28,47,76,120,185,280,416,608,878,1252,1765,2464,3408,4676,

%T 6364,8600,11545,15400,20424,26938,35346,46152,59981,77616,100016,

%U 128369,164140,209120,265510,335992,423840,533035,668404,835804,1042308,1296448

%N 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.

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

%H G. C. Greubel, <a href="/A261240/b261240.txt">Table of n, a(n) for n = 0..1000</a>

%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015.

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F 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.

%F 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.

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

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

%F a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(7/4) * sqrt(3) * n^(3/4)). - _Vaclav Kotesovec_, Oct 13 2015

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

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

%t 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}];

%t 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 *)

%o (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))};

%Y Cf. A058647, A186115, A186964, A187020.

%K nonn

%O 0,2

%A _Michael Somos_, Aug 12 2015