A229723 - OEIS

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

%S 1,1,0,0,-1,0,-2,-2,0,-2,2,0,0,0,0,4,-1,0,0,0,0,0,2,0,2,3,0,0,-2,0,0,

%T -2,0,-4,0,0,2,0,0,0,-2,0,-4,0,0,0,0,0,0,3,0,0,0,0,-2,-4,0,0,2,0,4,0,

%U 0,4,-1,0,0,0,0,0,4,0,0,2,0,0,0,0,0,-2,0,-2

%N Expansion of psi(q) * chi(-q^3) * phi(-q^6) in powers of q where phi(), psi(), chi() 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="/A229723/b229723.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 eta(q^2)^2 * eta(q^3) * eta(q^6) / (eta(q) * eta(q^12)) in powers of q.

%F Euler transform of period 12 sequence [ 1, -1, 0, -1, 1, -3, 1, -1, 0, -1, 1, -2, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = 13824^(1/2) (t / i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128583.

%F a(3*n + 2) = 0.

%e G.f. = 1 + q - q^4 - 2*q^6 - 2*q^7 - 2*q^9 + 2*q^10 + 4*q^15 - q^16 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ q^3, q^6] EllipticTheta[ 4, 0, q^6] EllipticTheta[ 2, 0, q^(1/2)] / (2 q^(1/8)), {q, 0, n}];

%t a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^2 QPochhammer[ q^3] QPochhammer[ q^6]/ (QPochhammer[ q] QPochhammer[ q^12]), {q, 0, n}];

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

%Y Cf. A128583.

%K sign

%O 0,7

%A _Michael Somos_, Sep 27 2013