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A138519 Expansion of q * (psi(q^5) / psi(q))^2 in powers of q where psi() is a Ramanujan theta function.

%I A138519 #17 Feb 16 2025 08:33:08
%S A138519 1,-2,3,-6,11,-16,24,-38,57,-82,117,-168,238,-328,448,-614,834,-1114,
%T A138519 1480,-1966,2592,-3384,4398,-5704,7361,-9436,12045,-15344,19470,
%U A138519 -24576,30922,-38822,48576,-60548,75259,-93342,115454,-142360,175104,-214958,263262
%N A138519 Expansion of q * (psi(q^5) / psi(q))^2 in powers of q where psi() is a Ramanujan theta function.
%C A138519 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700)
%H A138519 G. C. Greubel, <a href="/A138519/b138519.txt">Table of n, a(n) for n = 1..1000</a>
%H A138519 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A138519 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A138519 Expansion of ((eta(q^10) / eta(q^2))^2 * eta(q) / eta(q^5))^2 in powers of q.
%F A138519 Euler transform of period 10 sequence [ -2, 2, -2, 2, 0, 2, -2, 2, -2, 0, ...].
%F A138519 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u - v)^2 - v * (1 - u) * (1 - 5*u).
%F A138519 G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v)^4 - u * (1 - u) * (1 - 5*u) * v * (1 - v) * (1 - 5*v).
%F A138519 G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = (1/5) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A138518.
%F A138519 G.f.: x * (Product_{k>0} P(5, x^k) * P(10, x^k)^2)^2 where P(n, x) is the n-th cyclotomic polynomial.
%F A138519 a(n) = - A138520(n) unless n=0. -5 * a(n) = A138521(n) unless n=0.
%F A138519 Convolution inverse of A138516.
%F A138519 a(n) = -(-1)^n * A210458(n). - _Michael Somos_, Sep 16 2015
%F A138519 a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n/5)) / (2 * 5^(5/4) * n^(3/4)). - _Vaclav Kotesovec_, Nov 15 2017
%e A138519 G.f. = q - 2*q^2 + 3*q^3 - 6*q^4 + 11*q^5 - 16*q^6 + 24*q^7 - 38*q^8 + 57*q^9 + ...
%t A138519 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q^(5/2)] / EllipticTheta[ 2, 0, q^(1/2)])^2, {q, 0, n}]; (* _Michael Somos_, Sep 16 2015 *)
%o A138519 (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( ( eta(x + A) / eta(x^5 + A) * ( eta(x^10 + A) / eta(x^2 + A) )^2)^2, n))};
%Y A138519 Cf. A138516, A138520, A138521, A210458.
%K A138519 sign
%O A138519 1,2
%A A138519 _Michael Somos_, Mar 23 2008