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%I #11 Mar 12 2021 22:24:44
%S 1,2,4,7,12,20,32,50,76,113,166,240,342,482,672,928,1270,1724,2323,
%T 3108,4132,5460,7174,9376,12192,15780,20332,26086,33334,42432,53817,
%U 68018,85680,107584,134674,168092,209210,259680,321484,396996,489056,601052,737024
%N Expansion of psi(-x^3) / phi(-x) in powers of x where psi(), phi() are Ramanujan theta functions.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%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 q^(-3/8) * eta(q^2) * eta(q^3) * eta(q^12) / (eta(q)^2 * eta(q^6)) in powers of q.
%F Euler transform of period 12 sequence [ 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 0, ...].
%F G.f.: Product_{k>0} (1 + x^k) * (1 + x^k + x^(2*k)) * (1 + x^(6*k)).
%F a(n) ~ 5^(1/4) * exp(sqrt(5*n/6)*Pi) / (2^(11/4) * 3^(3/4) * n^(3/4)). - _Vaclav Kotesovec_, Oct 14 2015
%e G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 12*x^4 + 20*x^5 + 32*x^6 + 50*x^7 + 76*x^8 + ...
%e G.f. = q^3 + 2*q^11 + 4*q^19 + 7*q^27 + 12*q^35 + 20*q^43 + 32*q^51 + 50*q^59 + ...
%t nmax=60; CoefficientList[Series[Product[(1+x^k) * (1-x^(12*k))/( (1-x^k) * (1+x^(3*k))),{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Oct 14 2015 *)
%t a[ n_] := SeriesCoefficient[ 2^(-1/2) x^(-3/8) EllipticTheta[ 2, Pi/4, x^(3/2)] / EllipticTheta[ 4, 0, x], {x, 0, n}]; (* _Michael Somos_, Nov 01 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A)^2 * eta(x^6 + A)), n))};
%K nonn
%O 0,2
%A _Michael Somos_, Aug 13 2007
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