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A145722 Expansion of f(q) * f(q^5) / phi(-q^2)^2 in powers of q where f(), phi() are Ramanujan theta functions.

%I A145722 #21 Feb 16 2025 08:33:09
%S A145722 1,1,3,4,8,12,21,30,48,68,102,143,207,284,400,542,744,996,1344,1776,
%T A145722 2361,3088,4050,5248,6808,8742,11232,14310,18224,23052,29133,36601,
%U A145722 45936,57360,71528,88812,110110,135990,167704,206108,252912,309408
%N A145722 Expansion of f(q) * f(q^5) / phi(-q^2)^2 in powers of q where f(), phi() are Ramanujan theta functions.
%C A145722 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A145722 G. C. Greubel, <a href="/A145722/b145722.txt">Table of n, a(n) for n = 0..1000</a>
%H A145722 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 A145722 MathOverflow, <a href="http://mathoverflow.net/questions/230737/up-to-2000-a145722n-1-equiv-sigma4n-3-pmod5">Up to 2000...</a>, 2016.
%H A145722 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A145722 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A145722 Expansion of q^(-1/4) * eta(q^4) * eta(q^10)^3 / (eta(q) * eta(q^2) * eta(q^5) * eta(q^20)) in powers of q.
%F A145722 Euler transform of period 20 sequence [ 1, 2, 1, 1, 2, 2, 1, 1, 1, 0, 1, 1, 1, 2, 2, 1, 1, 2, 1, 0, ...].
%F A145722 G.f. is a period 1 Fourier series which satisfies f(-1 / (80 t)) = 20^(-1/2) g(t), where q = exp(2 Pi i t) and g() is the g.f. for A145723.
%F A145722 G.f.: Product_{k>0} (1 + x^(2*k)) * (1 - x^(10*k)) * (1 + x^(5*k)) / ((1 - x^k) * (1 + x^(10*k))).
%F A145722 a(n) = A036026(2*n).
%F A145722 a(n) ~ exp(2*Pi*sqrt(n/5)) / (4 * 5^(3/4) * n^(3/4)). - _Vaclav Kotesovec_, Oct 13 2015
%e A145722 G.f. = 1 + x + 3*x^2 + 4*x^3 + 8*x^4 + 12*x^5 + 21*x^6 + 30*x^7 + 48*x^8 + ...
%e A145722 G.f. = q + q^5 + 3*q^9 + 4*q^13 + 8*q^17 + 12*q^21 + 21*q^25 + 30*q^29 + ...
%t A145722 a[ n_] := SeriesCoefficient[ QPochhammer[ -x]  QPochhammer[ -x^5] / EllipticTheta[ 4, 0, x^2]^2, {x, 0, n}]; (* _Michael Somos_, Aug 26 2015 *)
%t A145722 nmax=60; CoefficientList[Series[Product[(1+x^(2*k)) * (1-x^(10*k)) * (1+x^(5*k)) / ((1-x^k) * (1 + x^(10*k))),{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Oct 13 2015 *)
%o A145722 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A) * eta(x^10 + A)^3 / (eta(x + A) * eta(x^2 + A) * eta(x^5 + A) * eta(x^20 + A)), n))};
%Y A145722 Cf. A036026.
%K A145722 nonn
%O A145722 0,3
%A A145722 _Michael Somos_, Oct 23 2008