A128640 - OEIS

%I #24 Mar 12 2021 22:24:44

%S 1,-4,10,-20,39,-76,140,-244,415,-696,1140,-1820,2861,-4448,6816,

%T -10292,15372,-22756,33356,-48408,69683,-99600,141312,-199036,278557,

%U -387608,536230,-737632,1009464,-1374888,1863764,-2514868,3378948,-4521672,6027000,-8002676

%N Expansion of (1/3) * (c(q^2)^2 / c(q)) / (b(q^2)^2 / b(q)) 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).

%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

%H Seiichi Manyama, <a href="/A128640/b128640.txt">Table of n, a(n) for n = 1..10000</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 q * (psi(q^3) / psi(q))^4 in powers of q where psi() is a Ramanujan theta function.

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

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

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

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

%F G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (81*u^2*v^2 + 9*u*v - 12*u + 30*u^2 - 108*u^2*v + 1) * v - u^3.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (1/9) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128637.

%F a(n) = - A128641(n) unless n = 0. Convolution inverse of A128633.

%F a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n/3)) / (2 * 3^(9/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 06 2018

%F Empirical: Sum_{n>=1} a(n)/exp(2*Pi*n) = 1/3 + (2/9)*sqrt(3) - (2/9)*sqrt(6)*3^(1/4). - _Simon Plouffe_, Mar 02 2021

%e G.f. = q - 4*q^2 + 10*q^3 - 20*q^4 + 39*q^5 - 76*q^6 + 140*q^7 - 244*q^8 + ...

%t a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q^(3/2)] / EllipticTheta[ 2, 0, q^(1/2)])^4, {q, 0, n}]; (* _Michael Somos_, Aug 26 2015 *)

%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( ((eta(x^6 + A) / eta(x^2 + A))^2 * eta(x + A) / eta(x^3 + A))^4, n))};

%Y Cf. A128633, A128641, A164617.

%K sign

%O 1,2

%A _Michael Somos_, Mar 16 2007