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A132972 Expansion of chi(q)^3 / chi(q^3) in powers of q where chi() is a Ramanujan theta function.

%I A132972 #31 Feb 16 2025 08:33:06
%S A132972 1,3,3,3,6,9,12,15,21,30,36,45,60,78,96,117,150,189,228,276,342,420,
%T A132972 504,603,732,885,1050,1245,1488,1773,2088,2454,2901,3420,3996,4662,
%U A132972 5460,6378,7404,8583,9972,11565,13344,15378,17748,20448,23472,26910,30876
%N A132972 Expansion of chi(q)^3 / chi(q^3) in powers of q where chi() is a Ramanujan theta function.
%C A132972 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A132972 G. C. Greubel, <a href="/A132972/b132972.txt">Table of n, a(n) for n = 0..1000</a>
%H A132972 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A132972 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A132972 Expansion of eta(q^2)^6 * eta(q^3) * eta(q^12) / (eta(q)^3 * eta(q^4)3 * eta(q^12)) in powers of q.
%F A132972 Euler transform of period 12 sequence [ 3, -3, 2, 0, 3, -2, 3, 0, 2, -3, 3, 0, ...].
%F A132972 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (2 + u*v) * (u*v - 1)^3 - (u - u^4) * (v - v^4).
%F A132972 G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u * (4 - 2*u + u^2) - v^3 * (1 + u + u^2).
%F A132972 G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = (2 + u1 * u2) - u3 * u6 * (1 + u1 + u2).
%F A132972 G.f. is a period 1 Fourier series which satisfies f(-1/(144*t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A062244.
%F A132972 G.f.: Product_{k>0} (1 + x^(2*k-1))^3 / (1 + x^(6*k-3)).
%F A132972 a(n) = 3 * A132975(n) unless n=0.
%F A132972 Empirical: Sum_{n>=1} exp(-Pi)^(n-1)*a(n) = (-2 + 2*sqrt(3))^(1/3). - Simon Plouffe, Feb 20 2011
%F A132972 a(n) ~ exp(2*Pi*sqrt(n)/3) / (2*sqrt(3)*n^(3/4)). - _Vaclav Kotesovec_, Sep 08 2015
%F A132972 It appears that the g.f. A(x) = F(x)^3, where F(x) = exp( Sum_{n >= 0} x^(3*n+1)/((3*n + 1)*(1 - (-1)^(n+1)*x^(3*n+1))) + x^(3*n+2)/((3*n + 2)*(1 - (-1)^n*x^(3*n + 2))) ). Cf. A273845. - _Peter Bala_, Dec 23 2021
%e A132972 G.f. = 1 + 3*q + 3*q^2 + 3*q^3 + 6*q^4 + 9*q^5 + 12*q^6 + 15*q^7 + 21*q^8 + ...
%t A132972 nmax = 60; CoefficientList[Series[Product[(1 + x^(2*k-1))^3 / (1 + x^(6*k-3)), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 08 2015 *)
%t A132972 a[ n_] := SeriesCoefficient[ QPochhammer[ -q, q^2]^3 / QPochhammer[ -q^3, q^6], {q, 0, n}]; (* _Michael Somos_, Oct 31 2015 *)
%o A132972 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^6 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^6 + A)^2), n))};
%Y A132972 Cf. A062244, A132975, A273845.
%K A132972 nonn
%O A132972 0,2
%A A132972 _Michael Somos_, Sep 06 2007