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A147699 Expansion of f(x) * f(x^5) / phi(-x^10)^2 in powers of x where f(), phi() are Ramanujan theta functions.

%I A147699 #14 Feb 16 2025 08:33:09
%S A147699 1,1,-1,0,0,0,1,-2,0,0,2,3,-5,0,0,2,4,-8,0,0,5,8,-14,0,0,6,12,-22,0,0,
%T A147699 13,21,-36,0,0,16,30,-54,0,0,28,48,-83,0,0,38,68,-120,0,0,60,102,-176,
%U A147699 0,0,80,143,-250,0,0,122,207,-356,0,0,162,284,-494,0,0
%N A147699 Expansion of f(x) * f(x^5) / phi(-x^10)^2 in powers of x where f(), phi() are Ramanujan theta functions.
%C A147699 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A147699 G. C. Greubel, <a href="/A147699/b147699.txt">Table of n, a(n) for n = 0..1000</a>
%H A147699 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A147699 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A147699 Expansion of q^(-1/4) * eta(q^2)^3 * eta(q^20) / (eta(q) * eta(q^4) * eta(q^5) * eta(q^10)) in powers of q.
%F A147699 Euler transform of period 20 sequence [ 1, -2, 1, -1, 2, -2, 1, -1, 1, 0, 1, -1, 1, -2, 2, -1, 1, -2, 1, 0, ...].
%F A147699 G.f. is a period 1 Fourier series which satisfies f(-1 / (80 t)) = (5/4)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A147702.
%F A147699 a(5*n + 2) = a(5*n + 3) = 0.
%F A147699 a(n) = A138532(2*n + 1). a(5*n + 1) = A145722(n).
%F A147699 a(5*n) = A261866(n). a(5*n + 2) = - A261526(n). - _Michael Somos_, Sep 03 2015
%e A147699 G.f. = 1 + x - x^2 + x^6 - 2*x^7 + 2*x^10 + 3*x^11 - 5*x^12 + 2*x^15 + ...
%e A147699 G.f. = q + q^5 - q^9 + q^25 - 2*q^29 + 2*q^41 + 3*q^45 - 5*q^49 + 2*q^61 + ...
%t A147699 a[ n_] := SeriesCoefficient[ QPochhammer[ -x] QPochhammer[ -x^5] / EllipticTheta[ 4, 0, x^10]^2, {x, 0, n}]; (* _Michael Somos_, Sep 02 2015 *)
%o A147699 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^20 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^5 + A) * eta(x^10 + A)), n))};
%Y A147699 Cf. A138532, A145722, A147702, A261526, A261866.
%K A147699 sign
%O A147699 0,8
%A A147699 _Michael Somos_, Nov 10 2008