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A134079 Expansion of q^(-2/3) * c(-q)^2 / 9 in powers of q where c(q) is a cubic AGM theta function.

%I A134079 #21 Feb 16 2025 08:33:06
%S A134079 1,-2,5,-4,8,-6,14,-8,14,-10,21,-16,20,-14,28,-16,31,-18,40,-20,32,
%T A134079 -28,42,-24,38,-32,62,-28,44,-30,56,-40,57,-34,70,-36,72,-38,70,-48,
%U A134079 62,-52,85,-44,68,-46,112,-56,74,-50,100,-64,80,-64,98,-56,108,-58,124
%N A134079 Expansion of q^(-2/3) * c(-q)^2 / 9 in powers of q where c(q) is a cubic AGM theta function.
%C A134079 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A134079 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A134079 G. C. Greubel, <a href="/A134079/b134079.txt">Table of n, a(n) for n = 0..10000</a>
%H A134079 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A134079 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A134079 Expansion of ( f(x^3)^3 / f(x) )^2 in powers of x where f() is a Ramanujan theta function.
%F A134079 Expansion of q^(-2/3) * eta(q)^2 * eta(q^4)^2 * eta(q^6)^18 / (eta(q^2) * eta(q^3)* eta(q^12))^6 in powers of q.
%F A134079 Euler transform of period 12 sequence [ -2, 4, 4, 2, -2, -8, -2, 2, 4, 4, -2, -4, ...].
%F A134079 G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (4/3) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263773.
%F A134079 a(n) = (-1)^n * A033686(n). 18 * a(n) = A134078(3*n + 2).
%F A134079 From _Michael Somos_, Feb 19 2015: (Start)
%F A134079 a(2*n + 1) = -2 * A098098(n).
%F A134079 Convolution square of A227696. (End)
%F A134079 Sum_{k=1..n} a(k) ~ (Pi^2/54) * n^2. - _Amiram Eldar_, Nov 23 2023
%e A134079 G.f. = 1 - 2*x + 5*x^2 - 4*x^3 + 8*x^4 - 6*x^5 + 14*x^6 - 8*x^7 + 14*x^8 - ...
%e A134079 G.f. = q^2 - 2*q^5 + 5*q^8 - 4*q^11 + 8*q^14 - 6*q^17 + 14*q^20 - 8*q^23 + ...
%t A134079 a[ n_] := SeriesCoefficient[ (QPochhammer[ -x^3]^3 / QPochhammer[ -x])^2, {x, 0, n}]; (* _Michael Somos_, Feb 19 2015 *)
%t A134079 a[ n_] := If[ n < 0, 0, (-1)^n DivisorSigma[ 1, 3 n + 2] / 3]; (* _Michael Somos_, Feb 19 2015 *)
%o A134079 (PARI) {a(n) = my(A); if( n<0, 0, A = x*O(x^n); polcoeff( ( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^9 / ( eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A) )^3 )^2, n))};
%o A134079 (PARI) {a(n) = if( n<0, 0, (-1)^n * sigma(3*n + 2) / 3)}; /* _Michael Somos_, Feb 19 2015 */
%Y A134079 Cf. A033686, A098098, A134078, A227696, A263773.
%Y A134079 Cf. A000700, A000122, A004016, A005882, A005928, A010054, A121373.
%K A134079 sign
%O A134079 0,2
%A A134079 _Michael Somos_, Oct 06 2007