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%I #17 Mar 12 2021 22:24:43
%S 1,0,1,2,2,2,4,4,6,8,9,12,16,18,22,28,33,40,50,58,70,84,98,116,138,
%T 160,188,222,256,298,348,400,463,536,614,706,812,926,1060,1212,1378,
%U 1568,1785,2022,2292,2598,2932,3312,3740,4208,4736,5328,5978,6708,7522,8416,9416
%N Expansion of psi(x^3)^2 / f(-x^2) in powers of x where psi(), f() are Ramanujan theta functions.
%C On page 63 of Watson 1936 is an equation with left side 2*rho(q) + omega(q) and the right side is 3 times the g.f. of this sequence. - _Michael Somos_, Jul 14 2015
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 50, Eq. (25.4).
%D George N. Watson, The final problem: an account of the mock theta functions, J. London Math. Soc., 11 (1936) 55-80.
%H G. C. Greubel, <a href="/A097196/b097196.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], 2015-2016.
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H George N. Watson, <a href="http://jlms.oxfordjournals.org/content/s1-11/1/55.extract">The final problem: an account of the mock theta functions</a>, J. London Math. Soc., 11 (1936) 55-80.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of q^(-2/3) * eta(x^6)^4 / (eta(x^2) * eta(x^3)^2) in powers of q. - _Michael Somos_, Jul 14 2015
%F G.f.: Product_{n >= 1} (1+q^(3*n))^4*(1-q^(3*n))^2/(1-q^(2*n)).
%F 3 * a(n) = A053253(n) + 2 * A053255(n). - _Michael Somos_, Jul 29 2015
%F a(n) ~ exp(Pi*sqrt(n/3)) / (12*sqrt(n)). - _Vaclav Kotesovec_, Oct 14 2015
%e G.f. = 1 + x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 4*x^6 + 4*x^7 + 6*x^8 + 8*x^9 + ...
%e G.f. = q^2 + q^8 + 2*q^11 + 2*q^14 + 2*q^17 + 4*q^20 + 4*q^23 + 6*q^26 + ...
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(3/2)]^2 / (4 x^(3/4) QPochhammer[ x^2]), {x, 0, n}]; (* _Michael Somos_, Jul 14 2015 *)
%t nmax=60; CoefficientList[Series[Product[(1+x^(3*k))^4 * (1-x^(3*k))^2 / (1-x^(2*k)),{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Oct 14 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^4 / (eta(x^2 + A) * eta(x^3 + A)^2), n))}; /* _Michael Somos_, Jul 14 2015 */
%Y Cf. A053253, A053255.
%K nonn
%O 0,4
%A _N. J. A. Sloane_, Sep 17 2004
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