%I #13 Mar 12 2021 22:24:44

%S 1,-3,1,3,-1,0,1,-6,0,6,-3,3,4,-12,1,12,-6,3,5,-24,1,24,-10,6,11,-42,

%T 4,42,-19,12,17,-72,4,69,-31,18,31,-120,9,114,-50,30,46,-189,11,180,

%U -79,48,77,-294,21,276,-122,72,112,-450,28,420,-183,108,173,-672

%N Expansion of f(-x, -x^5) * f(-x)^2 / f(-x^6)^3 in powers of x where f(, ) and f() are Ramanujan 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="/A132301/b132301.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 q^(1/3) * eta(q)^3 / (eta(q^2) * eta(q^3) * eta(q^6)) in powers of q.

%F Euler transform of period 6 sequence [ -3, -2, -2, -2, -3, 0, ...].

%F Given g.f. A(x), then B(q) = A(q^3) / (3*q) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (v^2 - 2*u)^3 - u^4 * (2*u - 3*v^2) * (4*u - 3*v^2).

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132302.

%F a(2*n) = A132179(n). a(2*n + 1) = -3 * A092848(n). - _Michael Somos_, Nov 01 2015

%e G.f. = 1 - 3*x + x^2 + 3*x^3 - x^4 + x^6 - 6*x^7 + 6*x^9 - 3*x^10 + 3*x^11 + ...

%e G.f. = 1/q - 3*q^2 + q^5 + 3*q^8 - q^11 + q^17 - 6*q^20 + 6*q^26 - 3*q^29 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^6] QPochhammer[ x^5, x^6] QPochhammer[ x]^2 / QPochhammer[ x^6]^2, {x, 0, n}]; (* _Michael Somos_, Nov 01 2015 *)

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

%Y Cf. A092848, A132179, A132302.

%K sign

%O 0,2

%A _Michael Somos_, Aug 17 2007