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A138522 Expansion of f(q, q^3)^2 / (f(q, q^4) * f(q^2, q^3)) in powers of q where f(, ) is the Ramanujan general theta function.

%I A138522 #14 Feb 16 2025 08:33:08
%S A138522 1,1,-1,0,-1,1,4,-4,-1,-3,3,12,-12,-2,-8,8,31,-30,-5,-20,19,72,-68,
%T A138522 -12,-44,41,154,-144,-24,-90,84,312,-289,-48,-178,164,603,-554,-92,
%U A138522 -336,307,1122,-1024,-168,-612,557,2024,-1836,-300,-1087,983,3552,-3206,-522
%N A138522 Expansion of f(q, q^3)^2 / (f(q, q^4) * f(q^2, q^3)) in powers of q where f(, ) is the Ramanujan general theta function.
%C A138522 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A138522 G. C. Greubel, <a href="/A138522/b138522.txt">Table of n, a(n) for n = 0..1000</a>
%H A138522 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A138522 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A138522 Expansion of (eta(q^2) / eta(q^5))^3 * eta(q^10) / eta(q) in powers of q.
%F A138522 Euler transform of period 10 sequence [ 1, -2, 1, -2, 4, -2, 1, -2, 1, 0, ...].
%F A138522 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u - v)^2 - u * (u + 4) * (1 - v).
%F A138522 G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u-v)^4 - u * (1 - u) * (4 + u) * v * (1 - v) * (4 + v).
%F A138522 G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = (5/4) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A138520.
%F A138522 G.f.: Product_{k>0} P(2, x^k)^4 * P(10, x^k) / P(5, x^k)^2 where P(n, x) is the n-th cyclotomic polynomial.
%F A138522 A095813(n) = a(n) unless n=0. Convolution inverse of A138520.
%e A138522 G.f. = 1 + q - q^2 - q^4 + q^5 + 4*q^6 - 4*q^7 - q^8 - 3*q^9 + 3*q^10 + 12*q^11 + ...
%t A138522 a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2] / QPochhammer[ q^5])^3 QPochhammer[ q^10] / QPochhammer[ q], {q, 0, n}]; (* _Michael Somos_, Sep 14 2015 *)
%o A138522 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^10 + A) / eta(x + A) * ( eta(x^2 + A) / eta(x^5 + A) )^3, n))};
%Y A138522 Cf. A095813, A138520.
%K A138522 sign
%O A138522 0,7
%A A138522 _Michael Somos_, Mar 23 2008