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A139631 Expansion of chi(x^5) / chi(-x^2) in powers of x where chi() is a Ramanujan theta function.

%I A139631 #16 Feb 16 2025 08:33:08
%S A139631 1,0,1,0,1,1,2,1,2,1,3,2,4,2,5,4,6,5,8,6,11,8,13,10,16,14,20,17,24,21,
%T A139631 31,26,37,32,44,41,54,49,64,59,79,72,94,86,111,106,132,126,156,149,
%U A139631 187,178,219,210,257,251,302,295,352,346,416,406,483,474,560
%N A139631 Expansion of chi(x^5) / chi(-x^2) in powers of x where chi() is a Ramanujan theta function.
%C A139631 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A139631 G. C. Greubel, <a href="/A139631/b139631.txt">Table of n, a(n) for n = 0..1000</a>
%H A139631 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A139631 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A139631 Expansion of q^(1/8) * eta(q^4) * eta(q^10)^2 / (eta(q^2) * eta(q^5) * eta(q^20)) in powers of q.
%F A139631 G.f. is a period 1 Fourier series which satisfies f(-1 / (640 t)) = 2^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A139632.
%F A139631 G.f.: Product_{k>0} (1 + x^(2*k)) * (1 + x^(5*k)) / (1 + x^(10*k)).
%F A139631 a(n) = A139632(2*n).
%F A139631 a(n) ~ exp(Pi*sqrt(n/5)) / (4 * 5^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 08 2015
%e A139631 G.f. = 1 + x^2 + x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + x^9 + 3*x^10 + 2*x^11 + ...
%e A139631 G.f. = 1/q + q^15 + q^31 + q^39 + 2*q^47 + q^55 + 2*q^63 + q^71 + 3*q^79 + ...
%t A139631 a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2, x^2] QPochhammer[ -x^5, x^10], {x, 0, n}]; (* _Michael Somos_, Sep 07 2015 *)
%t A139631 nmax = 40; CoefficientList[Series[Product[(1 + x^(2*k)) * (1 + x^(5*k)) / (1 + x^(10*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 08 2015 *)
%o A139631 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A) * eta(x^10 + A)^2 / (eta(x^2 + A) * eta(x^5 + A) * eta(x^20 + A)), n))};
%Y A139631 Cf. A139632.
%K A139631 nonn
%O A139631 0,7
%A A139631 _Michael Somos_, Apr 27 2008