A258092 - OEIS

%I #18 Mar 12 2021 22:24:48

%S 1,-2,4,-10,20,-36,64,-112,189,-308,492,-778,1210,-1844,2776,-4144,

%T 6114,-8914,12884,-18484,26302,-37124,52040,-72512,100415,-138196,

%U 189160,-257648,349184,-470932,632312,-845472,1125853,-1493222,1973060,-2597892,3408754

%N Expansion of f(x, x^2) / psi(x)^3 in powers of x where psi(), 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="/A258092/b258092.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)^2 * eta(q^3)^2 / (eta(q^2)^5 * eta(q^6)) in powers of q.

%F Euler transform of period 6 sequence [-2, 3, -4, 3, -2, 2, ...].

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

%F a(n) = A258093(3*n - 1).

%e G.f. = 1 - 2*x + 4*x^2 - 10*x^3 + 20*x^4 - 36*x^5 + 64*x^6 - 112*x^7 + ...

%e G.f. = 1/q - 2*q^2 + 4*q^5 - 10*q^8 + 20*q^11 - 36*q^14 + 64*q^17 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 QPochhammer[ x^3]^2 / ( QPochhammer[ x^2]^5 QPochhammer[ x^6]), {x, 0, n}]; (* _Michael Somos_, May 25 2015 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^3 + A)^2 / (eta(x^2 + A)^5 * eta(x^6 + A)), n))};

%Y Cf. A258093.

%K sign

%O 0,2

%A _Michael Somos_, May 19 2015