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A133563 Expansion of chi(-q) / chi(-q^5) in powers of q where chi() is a Ramanujan theta function.

%I A133563 #25 Feb 16 2025 08:33:06
%S A133563 1,-1,0,-1,1,0,0,-1,1,-1,2,-2,2,-2,2,-1,2,-3,2,-3,5,-5,4,-5,6,-4,4,-7,
%T A133563 7,-7,10,-11,10,-12,12,-10,12,-15,14,-16,22,-22,20,-24,26,-22,24,-30,
%U A133563 31,-33,40,-43,42,-46,48,-45,50,-58,58,-63,77,-79,76,-86,92,-86,92,-107,110,-116,134,-141,142,-154,160,-157
%N A133563 Expansion of chi(-q) / chi(-q^5) in powers of q where chi() is a Ramanujan theta function.
%C A133563 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A133563 In general, if m > 1 and g.f. = Product_{k>=1} (1 + x^(m*k))/(1 + x^k), then a(n) ~ (-1)^n * exp(Pi*sqrt((m+2)*n/(6*m))) * (m+2)^(1/4) / (4 * (6*m)^(1/4) * n^(3/4)) if m is even and a(n) ~ (-1)^n * exp(Pi*sqrt((m-1)*n/(6*m))) * (m-1)^(1/4) / (2^(3/2) * (6*m)^(1/4) * n^(3/4)) if m is odd. - _Vaclav Kotesovec_, Aug 31 2015
%C A133563 Denoted by t in Andrews and Berndt 2005. - _Michael Somos_, Apr 25 2016
%D A133563 G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part I, Springer, New York, 2005, MR2135178 (2005m:11001) See p. 337.
%H A133563 G. C. Greubel, <a href="/A133563/b133563.txt">Table of n, a(n) for n = 0..1000</a>
%H A133563 David J. Hemmer, <a href="https://arxiv.org/abs/2402.03643">Generating functions for fixed points of the Mullineux map</a>, arXiv:2402.03643 [math.CO], 2024.
%H A133563 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, p. 14.
%H A133563 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A133563 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A133563 Expansion of q^(-1/6) * eta(q) * eta(q^10) / ( eta(q^2) * eta(q^5) ) in powers of q.
%F A133563 Euler transform of period 10 sequence [ -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, ...].
%F A133563 G.f. is a period 1 Fourier series which satisfies f(-1 / (360 t)) = f(t) where q = exp(2 Pi i t).
%F A133563 Given g.f. A(x) then B(q) = q * A(q^6) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = v * (u^2 - v) + w^2 * (u^2 + v).
%F A133563 Given g.f. A(x) then B(q) = q * A(q^6) satisfies 0 = f(B(q), B(x^q), B(q^9)) where f(u, v, w) = (u^3 + w^3) * (v + v^3) + 2 * v^4 - v^2 + u^3 * w^3 * ( 2 - v^2 ).
%F A133563 Given g.f. A(x) then B(q) = q * A(q^6) satisfies 0 = f(B(q), B(q^2), B(q^5), B(q^10)) where f(u1, u2, u5, u10) = u1^2 * u5^2 + u1^2 * u10^4 + u1 * u2^2 * u5 * u10^2 + u2 * u5^2 * u10^3 + u2^3 * u10^3 - u2^2 * u10^2 - u1^3 * u5^3 - u1^4 * u10^2 - u1^3 * u2^2 * u5 - u1^2 * u2 * u5^2 * u10.
%F A133563 G.f.: Product_{k>0} P10(x^k) where P10 is the 10th cyclotomic polynomial.
%F A133563 G.f.: Product_{k>0} (1 + x^(5*k)) / (1 + x^k).
%F A133563 a(n) ~ (-1)^n * exp(Pi*sqrt(2*n/15)) / (2^(5/4) * 15^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Aug 31 2015
%e A133563 G.f. = 1 - x - x^3 + x^4 - x^7 + x^8 - x^9 + 2*x^10 - 2*x^11 - 2*x^13 + ...
%e A133563 G.f. = q - q^7 - q^19 + q^25 - q^43 + q^49 - q^55 + 2*q^61 - 2*q^67 + 2*q^73 - ...
%t A133563 a[ n_] := SeriesCoefficient[  QPochhammer[ x, x^2] / QPochhammer[ x^5, x^10], {x, 0, n}]; (* _Michael Somos_, Aug 26 2015 *)
%o A133563 (PARI) {a(n) = my(A); if( n<0, 0, A = x*O(x^n); polcoeff( eta(x + A) * eta(x^10 + A) / (eta(x^2 + A) * eta(x^5 + A)), n))};
%Y A133563 Cf. A081360 (m=2), A109389 (m=3), A261734 (m=4), A261736 (m=6), A113297 (m=7), A261735 (m=8), A261733 (m=9), A145707 (m=10).
%K A133563 sign
%O A133563 0,11
%A A133563 _Michael Somos_, Sep 16 2007