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A259529 Expansion of psi(-x^3)^2 / psi(-x) in powers of x where psi() is a Ramanujan theta function.

%I A259529 #17 Feb 16 2025 08:33:26
%S A259529 1,1,1,0,1,2,2,2,3,3,3,3,5,6,5,6,8,9,10,10,13,15,15,17,20,23,24,25,30,
%T A259529 34,36,39,45,50,53,57,65,73,77,83,94,104,110,118,132,145,154,166,185,
%U A259529 201,214,230,253,276,293,316,346,375,399,427,467,505,537,575
%N A259529 Expansion of psi(-x^3)^2 / psi(-x) in powers of x where psi() is a Ramanujan theta function.
%C A259529 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A259529 G. C. Greubel, <a href="/A259529/b259529.txt">Table of n, a(n) for n = 0..2500</a>
%H A259529 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 A259529 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A259529 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A259529 Expansion of f(x, x^5)^2 / f(x) in powers of x where f(,) is the general Ramanujan theta function.
%F A259529 Expansion of q^(-5/8) * eta(q^2) * eta(q^3)^2 * eta(q^12)^2 / (eta(q) * eta(q^4) * eta(q^6)^2) in powers of q.
%F A259529 Euler transform of period 12 sequence [ 1, 0, -1, 1, 1, 0, 1, 1, -1, 0, 1, -1, ...].
%F A259529 G.f. is a period 1 Fourier series which satisfies f(-1 / (768 t)) = (16/3)^1/2 (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A259538.
%F A259529 G.f.: Product_{k>0} (1 + x^k + x^(2*k)) * (1 - x^(3*k)) * (1 - x^(2*k) + x^(4*k)) * (1 + x^(6*k)).
%F A259529 a(n) ~ exp(Pi*sqrt(n/6)) / (6*sqrt(n)). - _Vaclav Kotesovec_, Jul 11 2016
%e A259529 G.f. = 1 + x + x^2 + x^4 + 2*x^5 + 2*x^6 + 2*x^7 + 3*x^8 + 3*x^9 + ...
%e A259529 G.f. = q^5 + q^13 + q^21 + q^37 + 2*q^45 + 2*q^53 + 2*q^61 + 3*q^69 + ...
%t A259529 a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{ -1, 0, 1, -1, -1, 0, -1, -1, 1, 0, -1, 1}[[Mod[k, 12, 1]]], {k, n}], {x, 0, n}];
%t A259529 a[ n_] := SeriesCoefficient[ (QPochhammer[ x^3, x^6] QPochhammer[ x^12])^2 / ( QPochhammer[ x, x^2] QPochhammer[ x^4]), {x, 0, n}];
%o A259529 (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 1, -1, 0, 1, -1, -1, 0, -1, -1, 1, 0, -1][k%12 + 1]), n))};
%o A259529 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n);polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^12 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^2), n))};
%Y A259529 Cf. A259538.
%K A259529 nonn
%O A259529 0,6
%A A259529 _Michael Somos_, Jun 29 2015