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A123861 Expansion of (f(q) * f(q^3) / (f(-q) * f(-q^3)))^2 in powers of q where f() is a Ramanujan theta function.

%I A123861 #24 Feb 16 2025 08:33:03
%S A123861 1,4,8,20,48,88,168,320,544,932,1584,2544,4080,6488,9984,15288,23232,
%T A123861 34568,51144,75152,108832,156736,224352,317728,447648,627292,871856,
%U A123861 1206068,1660416,2271032,3092976,4194464,5657728,7602096,10175760
%N A123861 Expansion of (f(q) * f(q^3) / (f(-q) * f(-q^3)))^2 in powers of q where f() is a Ramanujan theta function.
%C A123861 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A123861 G. C. Greubel, <a href="/A123861/b123861.txt">Table of n, a(n) for n = 0..1000</a>
%H A123861 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 A123861 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A123861 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A123861 Expansion of phi(q) * phi(q^3) / (phi(-q) * phi(-q^3)) in powers of q where phi() is a Ramanujan theta function. - _Michael Somos_, Aug 31 2014
%F A123861 Expansion of eta(q^2)^6 * eta(q^6)^6 / (eta(q)^4 * eta(q^3)^4 * eta(q^4)^2 * eta(q^12)^2) in powers of q. - _Michael Somos_, Aug 31 2014
%F A123861 Euler transform of period 12 sequence [4, -2, 8, 0, 4, -4, 4, 0, 8, -2, 4, 0, ...].
%F A123861 G.f. A(q) satisfies 0 = f(A(q), A(q^2)) where f(u, v) = (u - 1)^2 - 4 * u*v * (v - 1).
%F A123861 Let g.f. A(x) = u, then B(x) = u * (u-1) / 4, B(x^2) = ((u-1) / 4)^2 / u where B(x) is the g.f. for A123653.
%F A123861 G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (1/4) * g(t) where q = exp(2 Pi i t) and g() is the g.f. for A187197. - _Michael Somos_, Aug 31 2014
%F A123861 a(n) = 4 * A123647(n) unless n=0.
%F A123861 a(n) ~ exp(2*Pi*sqrt(n/3)) / (8 * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Oct 13 2015
%e A123861 G.f. = 1 + 4*q + 8*q^2 + 20*q^3 + 48*q^4 + 88*q^5 + 168*q^6 + 320*q^7 + ...
%t A123861 a[ n_] := SeriesCoefficient[ (QPochhammer[ -q] QPochhammer[ -q^3] / (QPochhammer[ q] QPochhammer[ q^3]))^2, {q, 0, n}]; (* _Michael Somos_, Aug 31 2014 *)
%t A123861 nmax=60; CoefficientList[Series[Product[(1+x^k)^6 * (1-x^k)^2 * (1+x^(3*k))^4 / ((1-x^(4*k))^2 * (1+x^(6*k))^2),{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Oct 13 2015 *)
%o A123861 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^6 * eta(x^6 + A)^6 / (eta(x + A)^4 * eta(x^3 + A)^4 * eta(x^4 + A)^2 * eta(x^12 + A)^2), n))};
%Y A123861 Cf. A123647, A187197.
%K A123861 nonn
%O A123861 0,2
%A A123861 _Michael Somos_, Oct 14 2006