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A261968 Expansion of phi(q^5) / phi(q) in powers of q where phi() is a Ramanujan theta function.

%I A261968 #12 Feb 16 2025 08:33:27
%S A261968 1,-2,4,-8,14,-22,36,-56,84,-126,184,-264,376,-528,732,-1008,1374,
%T A261968 -1856,2492,-3320,4394,-5784,7568,-9848,12756,-16442,21096,-26960,
%U A261968 34312,-43500,54956,-69184,86804,-108576,135392,-168336,208722,-258096,318320,-391632
%N A261968 Expansion of phi(q^5) / phi(q) in powers of q where phi() is a Ramanujan theta function.
%C A261968 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A261968 G. C. Greubel, <a href="/A261968/b261968.txt">Table of n, a(n) for n = 0..2500</a>
%H A261968 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A261968 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A261968 Expansion of eta(q)^2 * eta(q^4)^2 * eta(q^10)^5 / (eta(q^2)^5 * eta(q^5)^2 * eta(q^20)^2) in powers of q.
%F A261968 Euler transform of period 20 sequence [ -2, 3, -2, 1, 0, 3, -2, 1, -2, 0, -2, 1, -2, 3, 0, 1, -2, 3, -2, 0, ...].
%F A261968 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (1 - 2*u^2 + 5*u^4) * (1 - 2*v^2 + 5*v^4) - 4*(u^2 + 2*u*v - v^2)^2.
%F A261968 G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (v^2 + 3*u*v - u^2) * (u^2 + v^2) - u*v * (1 + 5*u^2*v^2).
%F A261968 G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = 5^(-1/2) * g(t) where q = exp(2 Pi i t) and g() is the g.f. for A144377.
%F A261968 G.f.: Product_{k>0} P(10, x^k)^3 * P(5, x^k) / P(20, x^k)^2 where P(n, x) is the n-th cyclotomic polynomial.
%F A261968 a(n) = (-1)^n * A138526(n). Convolution inverse is A144377.
%e A261968 G.f. = 1 - 2*q + 4*q^2 - 8*q^3 + 14*q^4 - 22*q^5 + 36*q^6 - 56*q^7 + ...
%t A261968 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^5] / EllipticTheta[ 3, 0, q], {q, 0, n}];
%o A261968 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^10 + A)^5 / (eta(x^2 + A)^5 * eta(x^5 + A)^2 * eta(x^20 + A)^2), n))};
%Y A261968 Cf. A138526, A144377.
%K A261968 sign
%O A261968 0,2
%A A261968 _Michael Somos_, Sep 06 2015