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%I #16 Mar 12 2021 22:24:46
%S 1,2,7,14,31,58,112,196,347,580,966,1554,2485,3872,5993,9102,13719,
%T 20384,30068,43836,63481,91048,129763,183448,257839,359862,499583,
%U 689312,946416,1292388,1756838,2376598,3201557,4293942,5736736,7633702,10121408,13370634
%N Expansion of chi(x)^2 / chi(-x^2)^6 in powers of x where chi() is a Ramanujan 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="/A224916/b224916.txt">Table of n, a(n) for n = 0..1000</a>
%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^(-5/12) * (eta(q^4)^2 / (eta(q) * eta(q^2)))^2 in powers of q.
%F Expansion of psi(x^2)^2 / f(-x)^2 = 1 / (chi(-x)^2 * chi(-x^2)^4) = 1 / (chi(x)^4 * chi(-x)^6 ) in powers of x where psi(), chi(), f() are Ramanujan theta functions.
%F Expansion of (chi(x)^4 - chi(-x)^4) / (8*x) in powers of x^2 where chi() is a Ramanujan theta function.
%F Euler transform of period 4 sequence [ 2, 4, 2, 0, ...].
%F G.f.: Product_{k>0} (1 + x^k)^2 * (1 + x^(2*k))^4.
%F G.f.: (Sum_{k>0} x^(k^2 - k)) / (Product_{k>0} (1 - x^k))^2. - _Michael Somos_, Jul 04 2013
%F a(n) = A112160(2*n + 1) / 4.
%F Convolution square of A098613. - _Michael Somos_, Jul 04 2013
%F a(n) ~ exp(2*Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2015
%e 1 + 2*x + 7*x^2 + 14*x^3 + 31*x^4 + 58*x^5 + 112*x^6 + 196*x^7 + 347*x^8 + ...
%e q^5 + 2*q^17 + 7*q^29 + 14*q^41 + 31*q^53 + 58*q^65 + 112*q^77 + 196*q^89 + ...
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q]^2 / (4 q^(1/2) QPochhammer[q]^2), {q, 0, n}]
%t a[ n_] := SeriesCoefficient[ 1 / QPochhammer[ q^2, q^4]^4 / QPochhammer[ q, q^2]^2, {q, 0, n}]
%t a[ n_] := SeriesCoefficient[ (QPochhammer[ -q, q^2]^4 - QPochhammer[ q, q^2]^4)/ 8, {q, 0, 2 n + 1}]
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A)^2 / (eta(x + A) * eta(x^2 + A)))^2, n))}
%Y Cf. A098613, A112160.
%K nonn
%O 0,2
%A _Michael Somos_, Apr 19 2013
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