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A145703 Expansion of chi(x) / chi(-x^10) in powers of x where chi() is a Ramanujan theta function.

%I A145703 #16 Feb 16 2025 08:33:09
%S A145703 1,1,0,1,1,1,1,1,2,2,3,3,3,4,4,5,6,6,7,8,10,11,11,13,15,17,18,20,23,
%T A145703 25,29,32,34,39,42,47,52,56,62,68,77,83,89,99,108,119,129,139,154,167,
%U A145703 183,199,214,234,253,276,299,322,350,378,413,445,476,518,559
%N A145703 Expansion of chi(x) / chi(-x^10) in powers of x where chi() is a Ramanujan theta function.
%C A145703 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A145703 G. C. Greubel, <a href="/A145703/b145703.txt">Table of n, a(n) for n = 0..1000</a>
%H A145703 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A145703 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A145703 Expansion of q^(-3/8) * eta(q^2)^2 * eta(q^20) / (eta(q) * eta(q^4) * eta(q^10) ) in powers of q.
%F A145703 Euler transform of period 20 sequence [ 1, -1, 1, 0, 1, -1, 1, 0, 1, 0, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, ...].
%F A145703 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 A145702.
%F A145703 G.f.: Product_{k>0} (1 + x^(2*k - 1)) / (1 - x^(20*k - 10)).
%F A145703 a(n) = (-1)^n * A145707(n) = A139632(2*n + 1).
%F A145703 a(n) ~ exp(Pi*sqrt(n/5)) / (4*5^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, Aug 30 2015
%e A145703 G.f. = 1 + x + x^3 + x^4 + x^5 + x^6 + x^7 + 2*x^8 + 2*x^9 + 3*x^10 + ...
%e A145703 G.f. = q^3 + q^11 + q^27 + q^35 + q^43 + q^51 + q^59 + 2*q^67 + 2*q^75 + ...
%t A145703 nmax = 100; CoefficientList[Series[Product[(1 + x^(2*k-1)) / (1 - x^(20*k-10)), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 30 2015 *)
%t A145703 a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ -x^10, x^10], {x, 0, n}]; (* _Michael Somos_, Sep 06 2015 *)
%o A145703 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^20 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^10 + A)), n))};
%Y A145703 Cf. A139632, A145702, A145707.
%K A145703 nonn
%O A145703 0,9
%A A145703 _Michael Somos_, Oct 17 2008