A263577 - OEIS

%I #9 Mar 12 2021 22:24:48

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

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

%U -2,0,0,0,4,0,0,0,0,0,2,0,0,6,0,-4,0,0,-2

%N Expansion of psi(-x^2) * psi(x^3)^2 / f(-x^24) 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="/A263577/b263577.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 phi(-x^12) * psi(-x^2) * chi(x^3)^2 in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

%F Expansion of eta(q^2) * eta(q^6)^4 * eta(q^8) / (eta(q^3)^2 * eta(q^4) * eta(q^24)) in powers of q.

%F Euler transform of period 24 sequence [ 0, -1, 2, 0, 0, -3, 0, -1, 2, -1, 0, -2, 0, -1, 2, -1, 0, -3, 0, 0, 2, -1, 0, -2, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 6^(3/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263571.

%F a(3*n) = A046113(n). a(3*n + 1) = 0. a(3*n + 2) = - A263548(n).

%e G.f. = 1 - x^2 + 2*x^3 - 2*x^5 - x^8 - 2*x^11 + 2*x^12 - 2*x^14 + 2*x^18 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, x] EllipticTheta[ 2, 0, x^(3/2)]^2 / (2^(5/2) x QPochhammer[ x^24]), {x, 0, n}];

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^12] EllipticTheta[ 2, Pi/4, x] / (2^(1/2) x^(1/4) QPochhammer[ x^3, -x^3]^2), {x, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^6 + A)^4 * eta(x^8 + A) / (eta(x^3 + A)^2 * eta(x^4 + A) * eta(x^24 + A)), n))};

%Y Cf. A046113, A263548, A263571.

%K sign

%O 0,4

%A _Michael Somos_, Oct 21 2015