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A138526 Expansion of phi(-q^5) / phi(-q) in powers of q where phi() is a Ramanujan theta function.

%I A138526 #18 Feb 16 2025 08:33:08
%S A138526 1,2,4,8,14,22,36,56,84,126,184,264,376,528,732,1008,1374,1856,2492,
%T A138526 3320,4394,5784,7568,9848,12756,16442,21096,26960,34312,43500,54956,
%U A138526 69184,86804,108576,135392,168336,208722,258096,318320,391632,480664
%N A138526 Expansion of phi(-q^5) / phi(-q) in powers of q where phi() is a Ramanujan theta function.
%C A138526 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A138526 G. C. Greubel, <a href="/A138526/b138526.txt">Table of n, a(n) for n = 0..1000</a>
%H A138526 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A138526 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A138526 Expansion of (eta(q^5) / eta(q))^2 * eta(q^2) / eta(q^10) in powers of q.
%F A138526 Euler transform of period 10 sequence [ 2, 1, 2, 1, 0, 1, 2, 1, 2, 0, ...].
%F A138526 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u^2 - v^2)^2 - u^2 * (v^2 - 1) * (5*v^2 - 1).
%F A138526 G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u^2 - v^2) * (u + v)^2 - u * v * (u^2 - 1) * (5*v^2 - 1).
%F A138526 G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u - v)^2 * w^2 - u * v * (v^2 - 1).
%F A138526 G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = (u1 * u6 - u2 * u3)^2 - u1 * u3 * (u2^2 - u6^2).
%F A138526 G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = 5^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A138532.
%F A138526 G.f.: Product_{k>0} P(5, x^k) / P(10, x^k) where P(n, x) is the n-th cyclotomic polynomial.
%F A138526 Convolution square is A138517. Convolution inverse is A138527.
%F A138526 a(n) ~ exp(2*Pi*sqrt(n/5)) / (2*5^(3/4)*n^(3/4)). - _Vaclav Kotesovec_, Sep 01 2015
%e A138526 G.f. = 1 + 2*q + 4*q^2 + 8*q^3 + 14*q^4 + 22*q^5 + 36*q^6 + 56*q^7 + 84*q^8 + ...
%t A138526 nmax=50; CoefficientList[Series[Product[(1+x^k)*(1-x^(5*k))/((1-x^k)*(1+x^(5*k))),{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Sep 01 2015 *)
%t A138526 a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^5] / EllipticTheta[ 4, 0, q], {q, 0, n}]; (* _Michael Somos_, Sep 14 2015 *)
%o A138526 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^5 + A) / eta(x + A))^2 * eta(x^2 + A) / eta(x^10 + A), n))};
%Y A138526 Cf. A138517, A138527, A138532.
%K A138526 nonn
%O A138526 0,2
%A A138526 _Michael Somos_, Mar 23 2008