A134078 - OEIS

%I #11 Mar 12 2021 22:24:45

%S 1,-6,18,-34,42,-36,30,-48,90,-118,108,-72,54,-84,144,-204,186,-108,

%T 66,-120,252,-272,216,-144,102,-186,252,-370,336,-180,180,-192,378,

%U -408,324,-288,90,-228,360,-476,540,-252,240,-264,504,-708,432,-288,198,-342

%N Expansion of (phi(-q) / phi(-q^2))^3 * phi(q^3)^5 / phi(-q^6) in powers of q where phi() is a Ramanujan theta function.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H G. C. Greubel, <a href="/A134078/b134078.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 Euler transform of period 12 sequence [ -6, 3, 4, 0, -6, -10, -6, 0, 4, 3, -6, -4, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 8 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A133739.

%F a(3*n + 2) = 18 * A134079(n). a(6*n + 5) = -36 * A098098(n).

%e G.f. = 1 - 6*x + 18*x^2 - 34*x^3 + 42*x^4 - 36*x^5 + 30*x^6 - 48*x^7 + 90*x^8 + ...

%t a[n_]:= SeriesCoefficient[(EllipticTheta[3, 0, -q]/EllipticTheta[3, 0, -q^2])^3*(EllipticTheta[3, 0, q^3]^5/EllipticTheta[3, 0, -q^6]), {q, 0, n}]; Table[a[n], {n,0,50}] (* _G. C. Greubel_, Jan 22 2018 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^6 * eta(x^4 + A)^3 * eta(x^6 + A)^23 / ( eta(x^2 + A)^9 * eta(x^3 + A)^10 * eta(x^12 + A)^9 ), n))};

%Y Cf. A098098, A133739, A134079.

%K sign

%O 0,2

%A _Michael Somos_, Oct 06 2007