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A132978 Expansion of q^(-2/3) * (psi(-q^3) / psi(-q)^3) * (c(q^2) / 3) in powers of q where psi() is a Ramanujan theta function and c() is a cubic AGM theta function.

%I A132978 #14 Feb 16 2025 08:33:06
%S A132978 1,3,7,15,32,63,114,201,350,591,967,1554,2468,3855,5916,8970,13471,
%T A132978 20007,29384,42771,61784,88530,125838,177642,249230,347484,481506,
%U A132978 663549,909788,1241127,1684824,2276781,3063657,4105275,5478698,7283709,9648360,12735471
%N A132978 Expansion of q^(-2/3) * (psi(-q^3) / psi(-q)^3) * (c(q^2) / 3) in powers of q where psi() is a Ramanujan theta function and c() is a cubic AGM theta function.
%C A132978 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A132978 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A132978 G. C. Greubel, <a href="/A132978/b132978.txt">Table of n, a(n) for n = 0..1000</a>
%H A132978 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 A132978 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A132978 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A132978 Expansion of q^(-2/3) * (psi(-q^3) / psi(-q)^3) * (c(q^2) / 3) in powers of q where psi() is a Ramanujan theta function and c() is a cubic AGM theta function.
%F A132978 Expansion of psi(-x^3)^3 * f(-x, x^2) / psi(-x)^4 in powers of x where psi(), f(,) are Ramanujan theta functions.
%F A132978 Expansion of q^(-2/3) * (eta(q^2) * eta(q^6))^2 * eta(q^3) * eta(q^12) / ( eta(q)* eta(q^4) )^3 in powers of q.
%F A132978 Euler transform of period 12 sequence [ 3, 1, 2, 4, 3, -2, 3, 4, 2, 1, 3, 0, ...].
%F A132978 a(n) = A132975(3*n + 2).
%F A132978 Convolution of A132974 and A045833.
%F A132978 a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(9/4) * n^(3/4)). - _Vaclav Kotesovec_, Oct 13 2015
%e A132978 G.f. = 1 + 3*x + 7*x^2 + 15*x^3 + 32*x^4 + 63*x^5 + 114*x^6 + 201*x^7 + ...
%e A132978 G.f. = q^2 + 3*q^5 + 7*q^8 + 15*q^11 + 32*q^14 + 63*q^17 + 114*q^20 + ...
%t A132978 a[ n_] := SeriesCoefficient[ 2^(1/2) x^(-5/8) EllipticTheta[ 3, 0, x^3]  QPochhammer[ x, -x] EllipticTheta[ 2, Pi/4, x^(3/2)]^3 / EllipticTheta[ 2, Pi/4, x^(1/2)]^4, {x, 0, n}] // Simplify;
%t A132978 nmax=60; CoefficientList[Series[Product[(1+x^(3*k))^3 * (1-x^(3*k))^4 * (1+x^(6*k)) / ( (1-x^k)^4 * (1+x^k) * (1+x^(2*k))^3),{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Oct 13 2015 *)
%o A132978 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^6 + A)^2 * eta(x^12 + A) / ( eta(x + A) * eta(x^4 + A))^3, n))};
%Y A132978 Cf. A045833, A132974, A132975.
%K A132978 nonn
%O A132978 0,2
%A A132978 _Michael Somos_, Sep 07 2007