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A132985 Expansion of chi(-q^5) / chi(-q)^5 in powers of q where chi() is a Ramanujan theta function.

%I A132985 #20 Feb 16 2025 08:33:06
%S A132985 1,5,15,40,95,205,420,820,1535,2785,4915,8460,14260,23590,38360,61440,
%T A132985 97055,151370,233355,355900,537395,803960,1192380,1754140,2560980,
%U A132985 3712205,5344570,7645600,10871080,15368350,21607220,30220360,42056415,58249680,80310510
%N A132985 Expansion of chi(-q^5) / chi(-q)^5 in powers of q where chi() is a Ramanujan theta function.
%C A132985 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A132985 G. C. Greubel, <a href="/A132985/b132985.txt">Table of n, a(n) for n = 0..1000</a>
%H A132985 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A132985 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A132985 Expansion of (eta(q^5) / eta(q^10)) / (eta(q) / eta(q^2))^5 in powers of q.
%F A132985 Euler transform of period 10 sequence [ 5, 0, 5, 0, 4, 0, 5, 0, 5, 0, ...].
%F A132985 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 + v + u * v * (2 - 4 * v).
%F A132985 G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v)^4 - u * v * (1 + 3 * u - 4 * u^2) * (1 + 3 * v - 4 * v^2).
%F A132985 G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = (1/4) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132980.
%F A132985 G.f.: Product_{k>0} (1 + x^k)^5 / (1 + x^(5*k)).
%F A132985 a(n) ~ exp(2*Pi*sqrt(2*n/5)) / (2^(11/4) * 5^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2015
%e A132985 G.f. = 1 + 5*q + 15*q^2 + 40*q^3 + 95*q^4 + 205*q^5 + 420*q^6 + 820*q^7 + ...
%t A132985 nmax = 40; CoefficientList[Series[Product[(1 + x^k)^5 / (1 + x^(5*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 07 2015 *)
%t A132985 a[ n_] := SeriesCoefficient[ QPochhammer[ q^5, q^10] / QPochhammer[ q, q^2]^5, {q, 0, n}]; (* _Michael Somos_, Oct 31 2015 *)
%o A132985 (PARI) {a(n) = my(A); if( n<0, 0, A = x*O(x^n); polcoeff( ( eta(x^5 + A) / eta(x^10 + A) ) / ( eta(x + A) / eta(x^2 + A) )^5, n))};
%Y A132985 Cf. A132980.
%K A132985 nonn
%O A132985 0,2
%A A132985 _Michael Somos_, Sep 07 2007
%E A132985 Typo in a(32) corrected by _G. C. Greubel_, Sep 28 2017