A257469 - OEIS

%I #9 Mar 12 2021 22:24:47

%S 1,-1,-1,0,0,1,1,0,-1,0,0,1,-1,1,0,-1,0,0,0,-1,-1,-1,1,1,0,1,1,0,1,0,

%T -1,0,1,-1,0,-1,1,-1,-1,0,0,0,0,1,1,0,-1,0,-1,0,0,0,0,-1,0,0,0,2,0,0,

%U 1,-1,0,1,0,1,0,1,0,1,-1,-1,-1,0,0,0,-2,-1,0

%N Expansion of f(-x) * psi(x^6) 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="/A257469/b257469.txt">Table of n, a(n) for n = 0..2500</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^(-19/24) * eta(q) * eta(q^12)^2 / eta(q^6) in powers of q.

%F Euler transform of period 12 sequence [ -1, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -2, ...].

%F -2 * a(n) = A246962(3*n + 2).

%e G.f. = 1 - x - x^2 + x^5 + x^6 - x^8 + x^11 - x^12 + x^13 - x^15 - x^19 + ...

%e G.f. = q^19 - q^43 - q^67 + q^139 + q^163 - q^211 + q^283 - q^307 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x] EllipticTheta[ 2, 0, x^3] / (2 x^(3/4)), {x, 0 ,n}];

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

%Y Cf. A246962.

%K sign

%O 0,58

%A _Michael Somos_, Apr 25 2015