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A210063 Expansion of psi(x^4) / phi(x) in powers of x where phi(), psi() are Ramanujan theta functions.

%I A210063 #19 Feb 16 2025 08:33:17
%S A210063 1,-2,4,-8,15,-26,44,-72,114,-178,272,-408,605,-884,1276,-1824,2580,
%T A210063 -3616,5028,-6936,9498,-12922,17468,-23472,31369,-41700,55156,-72616,
%U A210063 95172,-124202,161436,-209016,269616,-346562,443952,-566856,721530,-915642,1158608
%N A210063 Expansion of psi(x^4) / phi(x) in powers of x where phi(), psi() are Ramanujan theta functions.
%C A210063 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A210063 G. C. Greubel, <a href="/A210063/b210063.txt">Table of n, a(n) for n = 0..1000</a>
%H A210063 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A210063 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A210063 Expansion of q^(-1/2) * eta(q)^2 * eta(q^4) * eta(q^8)^2 / eta(q^2)^5 in powers of q.
%F A210063 Euler transform of period 8 sequence [ -2, 3, -2, 2, -2, 3, -2, 0, ...].
%F A210063 G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 8^(-1/2) * g(t) where q = exp(2 Pi i t) and g() is g.f. for A210030.
%F A210063 a(n) = (-1)^n * A187154(n). Convolution inverse of A208589.
%F A210063 a(n) ~ (-1)^n * exp(sqrt(n)*Pi) / (16*n^(3/4)). - _Vaclav Kotesovec_, Nov 17 2017
%e A210063 1 - 2*x + 4*x^2 - 8*x^3 + 15*x^4 - 26*x^5 + 44*x^6 - 72*x^7 + 114*x^8 + ...
%e A210063 q - 2*q^3 + 4*q^5 - 8*q^7 + 15*q^9 - 26*q^11 + 44*q^13 - 72*q^15 + 114*q^17 + ...
%t A210063 CoefficientList[Series[x^(-1/2) * EllipticTheta[2, 0, x^2] / (2*EllipticTheta[3, 0, x]), {x, 0, 50}], x] (* _Vaclav Kotesovec_, Nov 17 2017 *)
%o A210063 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A) * eta(x^8 + A)^2 / eta(x^2 + A)^5, n))}
%Y A210063 Cf. A187154, A208589, A210030.
%K A210063 sign
%O A210063 0,2
%A A210063 _Michael Somos_, Mar 16 2012