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A226034 Expansion of f(-x)^6 / (chi(x) * phi(-x)^6) in powers of x where phi(), chi(), f() are Ramanujan theta functions.

%I A226034 #23 Feb 16 2025 08:33:19
%S A226034 1,11,73,368,1552,5755,19337,60054,174801,481760,1266992,3198963,
%T A226034 7791921,18382187,42139440,94126547,205343040,438390320,917501570,
%U A226034 1885269635,3808353889,7571955531,14833349529,28657374307,54646711136,102932171227,191644299945
%N A226034 Expansion of f(-x)^6 / (chi(x) * phi(-x)^6) in powers of x where phi(), chi(), f() are Ramanujan theta functions.
%C A226034 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A226034 G. C. Greubel, <a href="/A226034/b226034.txt">Table of n, a(n) for n = 0..1000</a>
%H A226034 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 A226034 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A226034 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A226034 Expansion of q^(-19/24) * eta(q^2)^4 * eta(q^3)^6 * eta(q^4) / eta(q)^11 in powers of q.
%F A226034 a(n) = 1/12 * A001935(9*n + 7).
%F A226034 a(n) ~ exp(3*Pi*sqrt(n/2)) / (2^(19/4) * 3^(5/2) * n^(3/4)). - _Vaclav Kotesovec_, Oct 14 2015
%e A226034 1 + 11*x + 73*x^2 + 368*x^3 + 1552*x^4 + 5755*x^5 + 19337*x^6 + 60054*x^7 + ...
%e A226034 q^19 + 11*q^43 + 73*q^67 + 368*q^91 + 1552*q^115 + 5755*q^139 + 19337*q^163 + ...
%t A226034 nmax=60; CoefficientList[Series[Product[(1+x^k)^4 * (1-x^(3*k))^6 * (1-x^(4*k)) / (1-x^k)^7,{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Oct 14 2015 *)
%t A226034 eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[q^(-19/24)* eta[q^2]^4*eta[q^3]^6*eta[q^4]/eta[q]^11, {q, 0, n}]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Mar 15 2018 *)
%o A226034 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^3 + A)^6 * eta(x^4 + A) / eta(x + A)^11, n))}
%Y A226034 Cf. A001935.
%K A226034 nonn
%O A226034 0,2
%A A226034 _Michael Somos_, May 28 2013