cp's OEIS Frontend

A132979 Expansion of psi(q^3) / psi(q)^3 in powers of q where psi() is a Ramanujan theta function.

%I A132979 #12 Feb 16 2025 08:33:06
%S A132979 1,-3,6,-12,24,-45,78,-132,222,-363,576,-900,1392,-2121,3180,-4716,
%T A132979 6936,-10098,14550,-20796,29520,-41595,58176,-80856,111750,-153561,
%U A132979 209820,-285240,385968,-519840,696960,-930516,1237470,-1639314,2163456,-2845080,3728904,-4871211
%N A132979 Expansion of psi(q^3) / psi(q)^3 in powers of q where psi() is a Ramanujan theta function.
%C A132979 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A132979 G. C. Greubel, <a href="/A132979/b132979.txt">Table of n, a(n) for n = 0..1000</a>
%H A132979 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A132979 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A132979 Expansion of eta(q)^3 * eta(q^6)^2 / ( eta(q^2)^6 * eta(q^3) ) in powers of q.
%F A132979 Euler transform of period 6 sequence [ -3, 3, -2, 3, -3, 2, ...].
%F A132979 G.f.: Product_{k>0} (1 + x^(3*k)) * (1 - x^(6*k)) / ( (1 + x^k) * (1 - x^(2*k)) )^3.
%F A132979 a(n) = (-1)^n * A132974(n). Convolution inverse of A107760.
%e A132979 G.f. = 1 - 3*q + 6*q^2 - 12*q^3 + 24*q^4 - 45*q^5 + 78*q^6 - 132*q^7 + ...
%t A132979 a[ n_] := SeriesCoefficient[ 4 EllipticTheta[ 2, 0, q^(3/2)] / EllipticTheta[ 2, 0, q^(1/2)]^3, {q, 0, n}]; (* _Michael Somos_, Jun 20 2015 *)
%o A132979 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^6 + A )^2 / ( eta(x^2 + A)^6 * eta(x^3 + A) ), n))};
%Y A132979 Cf. A107760, A132974.
%K A132979 sign
%O A132979 0,2
%A A132979 _Michael Somos_, Sep 07 2007