A128637 - OEIS

%I #28 Mar 11 2021 20:40:02

%S 1,-8,24,-24,-40,144,-120,-192,600,-456,-688,2016,-1464,-2096,5952,

%T -4176,-5800,15984,-10920,-14816,39888,-26688,-35488,93888,-61752,

%U -80824,210576,-136536,-176320,453456,-290448,-370688,942936,-597600,-755024,1901952,-1194216

%N Expansion of 3 * (b(q)^2/b(q^2)) / (c(q)^2/c(q^2)) in powers of q where b(), c() are cubic AGM theta functions.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H G. C. Greubel, <a href="/A128637/b128637.txt">Table of n, a(n) for n = 0..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of (phi(-q) / phi(-q^3))^4 in powers of q where phi() is a Ramanujan theta function.

%F Expansion of ((eta(q) / eta(q^3))^2 * (eta(q^6) / eta(q^2)))^4 in powers of q.

%F Euler transform of period 6 sequence [ -8, -4, 0, -4, -8, 0, ...].

%F G.f.: A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u * (1-v) * (9-v) - (u-v)^2.

%F G.f.: A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u^2*v^2 + 9*u*v - 12*u*v^2 + 30*v^2 - 108*v + 81) * u - v^3.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 9 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128640.

%F G.f.: (Product_{k>0} (1 - x^k + x^(2*k)) / (1 + x^k + x^(2*k)) )^4.

%F a(n) = -8 * A123633(n) unless n=0. Convolution inverse of A128639.

%F Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = 9 + 3*sqrt(3) - 3*sqrt(9+6*sqrt(3)). - _Simon Plouffe_, Mar 02 2021

%e G.f. = 1 - 8*q + 24*q^2 - 24*q^3 - 40*q^4 + 144*q^5 - 120*q^6 - 192*q^7 + ...

%t a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q] / EllipticTheta[ 4, 0, q^3])^4, {q, 0, n}]; (* _Michael Somos_, Apr 06 2013 *)

%t a[ n_] := SeriesCoefficient[ (QPochhammer[ -q^3, q^3] / QPochhammer[ -q, q])^4 /(QPochhammer[ q^3] / QPochhammer[ q])^4, {q, 0, n}]; (* _Michael Somos_, Apr 06 2013 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ((eta(x + A) / eta(x^3 + A))^2 * eta(x^6 + A) / eta(x^2 + A))^4, n))};

%Y Cf. A123633, A128639, A128640.

%K sign

%O 0,2

%A _Michael Somos_, Mar 16 2007