cp's OEIS Frontend

A134078 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.

%I A134078 #12 Feb 16 2025 08:33:06
%S A134078 1,-6,18,-34,42,-36,30,-48,90,-118,108,-72,54,-84,144,-204,186,-108,
%T A134078 66,-120,252,-272,216,-144,102,-186,252,-370,336,-180,180,-192,378,
%U A134078 -408,324,-288,90,-228,360,-476,540,-252,240,-264,504,-708,432,-288,198,-342
%N A134078 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 A134078 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A134078 G. C. Greubel, <a href="/A134078/b134078.txt">Table of n, a(n) for n = 0..1000</a>
%H A134078 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A134078 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A134078 Euler transform of period 12 sequence [ -6, 3, 4, 0, -6, -10, -6, 0, 4, 3, -6, -4, ...].
%F A134078 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 A134078 a(3*n + 2) = 18 * A134079(n). a(6*n + 5) = -36 * A098098(n).
%e A134078 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 A134078 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 A134078 (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 A134078 Cf. A098098, A133739, A134079.
%K A134078 sign
%O A134078 0,2
%A A134078 _Michael Somos_, Oct 06 2007