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A101195 Expansion of psi(x^3) / psi(x) in powers of x where psi() is a Ramanujan theta function.

%I A101195 #16 Feb 16 2025 08:32:55
%S A101195 1,-1,1,-1,2,-3,3,-4,6,-7,8,-10,13,-16,18,-22,28,-33,38,-45,55,-65,74,
%T A101195 -87,104,-121,138,-160,188,-217,247,-284,330,-378,428,-489,562,-640,
%U A101195 722,-820,936,-1059,1191,-1345,1524,-1717,1924,-2163,2438,-2734,3054,-3419
%N A101195 Expansion of psi(x^3) / psi(x) in powers of x where psi() is a Ramanujan theta function.
%C A101195 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A101195 G. C. Greubel, <a href="/A101195/b101195.txt">Table of n, a(n) for n = 0..1000</a>
%H A101195 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A101195 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A101195 Expansion of q^(-1/2) * theta_2(q^3) / theta_2(q) in powers of q^2.
%F A101195 Expansion of q^(-1/4) * eta(q) * eta(q^6)^2 / (eta(q^2)^2 * eta(q^3)) in powers of q.
%F A101195 Given g.f. A(x), B(q) = A(q^4) * q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = u^3 - v + 3*u*v^2 - 3*u^2*v^3.
%F A101195 Given g.f. A(x), B(q) = (A(q^4) * q)^2 satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^2 - v + v^2 + 3*u^2*v.
%F A101195 a(n) = (-1)^n * A036018(n).
%F A101195 G.f.: Product_{k>0} (1 + x^k + x^(2*k)) * (1 - x^k + x^(2*k))^2.
%F A101195 a(n) ~ (-1)^n * exp(Pi*sqrt(n/3)) / (2^(3/2) * 3^(3/4) * n^(3/4)). - _Vaclav Kotesovec_, Nov 16 2017
%e A101195 G.f. = 1 - x + x^2 - x^3 + 2*x^4 - 3*x^5 + 3*x^6 - 4*x^7 + 6*x^8 + ...
%e A101195 G.f. = q - q^5 + q^9 - q^13 + 2*q^17 - 3*q^21 + 3*q^25 - 4*q^29 + 6*q^33 + ...
%t A101195 a[ n_] := SeriesCoefficient[ x^(-1/4) EllipticTheta[ 2, 0, x^(3/2)] / EllipticTheta[ 2, 0, x^(1/2)], {x, 0, n}];
%o A101195 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^2 / (eta(x^2 + A)^2 * eta(x^3 + A)), n))};
%Y A101195 Cf. A036018.
%K A101195 sign
%O A101195 0,5
%A A101195 _Michael Somos_, Dec 03 2004