A257400 - OEIS

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

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

%T 0,0,2,4,-2,0,0,0,4,0,0,-4,0,2,4,0,0,0,-2,1,2,4,0,0,2,0,0,4,0,-4,0,0,

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

%N Expansion of psi(q) * phi(q^2) * chi(-q^3) 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="/A257400/b257400.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 eta(q^3) * eta(q^4)^5 / (eta(q) * eta(q^8)^2 * eta(q^6)) in powers of q.

%F Euler transform of period 24 sequence [1, 1, 0, -4, 1, 1, 1, -2, 0, 1, 1, -4, 1, 1, 0, -2, 1, 1, 1, -4, 0, 1, 1, -2, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 2592^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A257399.

%F a(8*n + 7) = 0.

%e G.f. = 1 + q + 2*q^2 + 2*q^3 - q^4 - 2*q^6 + 2*q^8 - 4*q^11 - 2*q^12 + ...

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

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^4 + A)^5 / (eta(x + A) * eta(x^8 + A)^2 * eta(x^6 + A)), n))};

%Y Cf. A257399.

%K sign

%O 0,3

%A _Michael Somos_, Apr 21 2015