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A212318 Expansion of phi(q^2)^2 / (phi(-q) * phi(q^4)) in powers of q where phi() is a Ramanujan theta function.

%I A212318 #18 Feb 16 2025 08:33:17
%S A212318 1,2,8,16,32,60,96,160,256,394,624,944,1408,2092,3008,4320,6144,8612,
%T A212318 12072,16720,22976,31424,42528,57312,76800,102254,135728,179104,
%U A212318 235264,307852,400704,519808,671744,864672,1109904,1419456,1809568,2300284,2914272,3682400
%N A212318 Expansion of phi(q^2)^2 / (phi(-q) * phi(q^4)) in powers of q where phi() is a Ramanujan theta function.
%C A212318 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A212318 G. C. Greubel, <a href="/A212318/b212318.txt">Table of n, a(n) for n = 0..1000</a>
%H A212318 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A212318 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A212318 Expansion of chi(q^2)^5 / (chi(-q) * chi(q^4))^2 in powers of q where chi() is a Ramanujan theta function.
%F A212318 Expansion of eta(q^4)^12 * eta(q^16)^2 / (eta(q)^2 * eta(q^2)^3 * eta(q^8)^9) in powers of q.
%F A212318 Euler transform of period 16 sequence [ 2, 5, 2, -7, 2, 5, 2, 2, 2, 5, 2, -7, 2, 5, 2, 0, ...].
%F A212318 a(n) = 2 * A215348(n) unless n=0. a(2*n) = A014969(n). a(2*n + 1) = 2 * A232772(n).
%F A212318 a(n) ~ exp(sqrt(n)*Pi)/(4*sqrt(2)*n^(3/4)). - _Vaclav Kotesovec_, Sep 08 2017
%e A212318 G.f. = 1 + 2*q + 8*q^2 + 16*q^3 + 32*q^4 + 60*q^5 + 96*q^6 + 160*q^7 + ...
%t A212318 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^2]^2 / (EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^4]), {q, 0, n}]
%o A212318 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^12 * eta(x^16 + A)^2 / (eta(x + A)^2 * eta(x^2 + A)^3 * eta(x^8 + A)^9), n))}
%Y A212318 Cf. A014969, A215348, A232772.
%K A212318 nonn
%O A212318 0,2
%A A212318 _Michael Somos_, Oct 25 2013