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A263073 Expansion of phi(-x^5) / (chi(-x) * chi(-x^15)) in powers of x where phi(), chi() are Ramanujan theta functions.

%I A263073 #15 Feb 16 2025 08:33:27
%S A263073 1,1,1,2,2,1,2,3,2,4,4,4,5,6,6,8,9,9,12,12,13,16,18,18,22,24,25,29,32,
%T A263073 34,40,43,45,52,56,60,68,74,78,88,95,101,113,122,130,145,156,166,184,
%U A263073 198,209,231,249,264,290,311,331,361,388,412,448,480,510,554
%N A263073 Expansion of phi(-x^5) / (chi(-x) * chi(-x^15)) in powers of x where phi(), chi() are Ramanujan theta functions.
%C A263073 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A263073 G. C. Greubel, <a href="/A263073/b263073.txt">Table of n, a(n) for n = 0..2500</a>
%H A263073 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 A263073 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A263073 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A263073 Expansion of q^(-2/3) * eta(q^2) * eta(q^5)^2 * eta(q^30) / (eta(q) * eta(q^10) * eta(q^15)) in powers of q.
%F A263073 Euler transform of period 30 sequence [1, 0, 1, 0, -1, 0, 1, 0, 1, -1, 1, 0, 1, 0, 0, 0, 1, 0, 1, -1, 1, 0, 1, 0, -1, 0, 1, 0, 1, -1, ...].
%F A263073 a(n) ~ exp(sqrt(7*n/5)*Pi/3) / (2*sqrt(5*n)). - _Vaclav Kotesovec_, Jul 11 2016
%e A263073 G.f. = 1 + x + x^2 + 2*x^3 + 2*x^4 + x^5 + 2*x^6 + 3*x^7 + 2*x^8 + 4*x^9 + ...
%e A263073 G.f. = q^2 + q^5 + q^8 + 2*q^11 + 2*q^14 + q^17 + 2*q^20 + 3*q^23 + 2*q^26 + ...
%t A263073 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^5] / (QPochhammer[ x, x^2] QPochhammer[ x^15, x^30]), {x, 0, n}];
%t A263073 nmax = 100; CoefficientList[Series[Product[(1+x^k) * (1-x^(5*k)) * (1+x^(15*k)) / (1+x^(5*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jul 11 2016 *)
%o A263073 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^5 + A)^2 * eta(x^30 + A) / (eta(x + A) * eta(x^10 + A) * eta(x^15 + A)), n))};
%o A263073 (PARI) q='q+O('q^99); Vec(eta(q^2)*eta(q^5)^2*eta(q^30)/(eta(q)*eta(q^10)*eta(q^15))) \\ _Altug Alkan_, Jul 31 2018
%K A263073 nonn
%O A263073 0,4
%A A263073 _Michael Somos_, Oct 08 2015