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A213267 Expansion of phi(q^9) / (psi(-q) * chi(q^3)) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.

%I A213267 #19 Feb 16 2025 08:33:17
%S A213267 1,1,1,1,2,3,4,5,7,10,12,15,20,26,32,39,50,63,76,92,114,140,168,201,
%T A213267 244,295,350,415,496,591,696,818,967,1140,1332,1554,1820,2126,2468,
%U A213267 2861,3324,3855,4448,5126,5916,6816,7824,8970,10292,11793,13471,15372,17548
%N A213267 Expansion of phi(q^9) / (psi(-q) * chi(q^3)) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
%C A213267 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A213267 G. C. Greubel, <a href="/A213267/b213267.txt">Table of n, a(n) for n = 0..1000</a>
%H A213267 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 A213267 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A213267 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A213267 Expansion of eta(q^2) * eta(q^3) * eta(q^12) * eta(q^18)^5 / (eta(q) * eta(q^4) * eta(q^6)^2 * eta(q^9)^2 * eta(q^36)^2) in powers of q.
%F A213267 Euler transform of period 36 sequence [ 1, 0, 0, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 0, 0, 1, 1, -2, 1, 1, 0, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 0, 0, 1, 0, ...].
%F A213267 a(n) = A132975(n) unless n=0.
%F A213267 a(2*n) = A128129(n). a(2*n + 1) = A132302.
%F A213267 a(3*n) = A164617(n). a(3*n + 1) = A132977(n). a(3*n + 2) = A132978(n).
%F A213267 a(n) ~ exp(2*Pi*sqrt(n)/3) / (2 * 3^(3/2) * n^(3/4)). - _Vaclav Kotesovec_, Oct 14 2015
%e A213267 1 + q + q^2 + q^3 + 2*q^4 + 3*q^5 + 4*q^6 + 5*q^7 + 7*q^8 + 10*q^9 + ...
%t A213267 nmax=60; CoefficientList[Series[Product[(1+x^k) * (1+x^(6*k)) * (1+x^(9*k))^5 * (1-x^(9*k))^3 / ((1-x^(4*k)) * (1+x^(3*k)) * (1-x^(36*k))^2),{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Oct 14 2015 *)
%o A213267 (PARI) {a(n) = local(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^18 + A)^5 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^2 * eta(x^9 + A)^2 * eta(x^36 + A)^2), n))}
%Y A213267 Cf. A128129, A132302, A132975, A132977, A132978, A164617.
%K A213267 nonn
%O A213267 0,5
%A A213267 _Michael Somos_, Jun 07 2012