A294387 - OEIS

%I #26 May 07 2023 07:37:52

%S 1,-3,6,-12,21,-36,60,-96,150,-228,342,-504,732,-1050,1488,-2088,2901,

%T -3996,5460,-7404,9972,-13344,17748,-23472,30876,-40413,52644,-68268,

%U 88152,-113364,145224,-185352,235734,-298800,377514,-475488,597108,-747690,933672

%N Expansion of chi(q^3) / chi^3(q) in powers of q where chi() is a Ramanujan theta function.

%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 G. C. Greubel, <a href="/A294387/b294387.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 eta(q)^3 * eta(q^4)^3 * eta(q^6)^2 / (eta(q^2)^6 * eta(q^3) * eta(q^12)) in powers of q.

%F Expansion of (c(q) - c(q^4)) * (c(q) - 4*c(q^4)) / (c(q) + 2*c(q^4))^2 in powers of q where c(q) is a cubic AGM theta function.

%F Expansion of b(q^2) / b(-q) = b(q^2) / (2*b(q^4) - b(q)) in powers of q where b() is a cubic AGM theta function.

%F Expansion of (3*a(q^12) - a(q^4)) / (a(q) + a(q^2)) = -1/2 + 3/2*(a(-q^3) + 2*a(q^3)) / (2*a(q) + a(-q)) in powers of q where a() is a cubic AGM theta function.

%F Euler transform of period 12 sequence [-3, 3, -2, 0, -3, 2, -3, 0, -2, 3, -3, 0, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128111.

%F G.f. A(q) = (1 - T(q)) / (1 + 2*T(q))  where T(q) = q*A128111(q^3).

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u*v) + 3*(u*v)^2 - 4*(u*v)^3 + 2*(u*v)^4 - (u^3 + v^3).

%F G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u*(1 + u + u^2) - v^3*(1 - 2*u + 4*u^2).

%F 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 + u2 + u1*u2 - u3*u6 - 2*u1*u2*u3*u6.

%F G.f.: Product_{k>0} (1 + x^(6*k-3)) / (1 + x^(2*k-1))^3.

%F a(n) = (-1)^n * A128128(n). Convolution inverse of A132972.

%F a(3*n + 1) = -3 * A164270(n). a(3*n + 2) = 6 * A164271(n).

%F Empirical : Sum_{n>=0} a(n)/exp(Pi*n) = 1/2*(2+2*3^(1/2))^(1/3), validated up to 1000 digits. - _Simon Plouffe_, May 06, 2023

%e G.f. = 1 - 3*x + 6*x^2 - 12*x^3 + 21*x^4 - 36*x^5 + 60*x^6 - 96*x^7 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ q, -q]^3 / QPochhammer[ q^3, -q^3], {q, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^6 + A)^2 / (eta(x^2 + A)^6 * eta(x^3 + A) * eta(x^12 + A)), n))};

%o (PARI) lista(nn) = {q='q+O('q^nn); Vec(eta(q)^3*eta(q^4)^3*eta(q^6)^2/(eta(q^2)^6*eta(q^3)*eta(q^12)))} \\ _Altug Alkan_, Mar 21 2018

%Y Cf. A128111, A128128, A132972, A164270, A164271.

%K sign

%O 0,2

%A _Michael Somos_, Oct 29 2017