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A132002 Expansion of phi(q^3) / phi(q) in powers of q where phi() is a Ramanujan theta function.

%I A132002 #23 Jul 19 2025 13:06:47
%S A132002 1,-2,4,-6,10,-16,24,-36,52,-74,104,-144,198,-268,360,-480,634,-832,
%T A132002 1084,-1404,1808,-2316,2952,-3744,4728,-5946,7448,-9294,11556,-14320,
%U A132002 17688,-21780,26740,-32736,39968,-48672,59122,-71644,86616,-104484,125768,-151072
%N A132002 Expansion of phi(q^3) / phi(q) in powers of q where phi() is a Ramanujan theta function.
%C A132002 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A132002 G. C. Greubel, <a href="/A132002/b132002.txt">Table of n, a(n) for n = 0..1000</a>
%H A132002 Shane Chern, Dennis Eichhorn, Shishuo Fu, and James A. Sellers, <a href="https://arxiv.org/abs/2507.10965">Convolutive sequences, I: Through the lens of integer partition functions</a>, arXiv:2507.10965 [math.CO], 2025. See p. 15.
%H A132002 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A132002 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A132002 Expansion of eta(q)^2 * eta(q^4)^2 * eta(q^6)^5 / ( eta(q^2)^5 * eta(q^3)^2 * eta(q^12)^2) in powers of q.
%F A132002 Euler transform of period 12 sequence [ -2, 3, 0, 1, -2, 0, -2, 1, 0, 3, -2, 0, ...].
%F A132002 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (v+u) * (v-u) + (1 - u*v) * (1 - 3*u*v).
%F A132002 G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u^3 - v + 3*u*v^2 * (1 - u*v).
%F A132002 G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 3^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A139007. - _Michael Somos_, Apr 04 2015
%F A132002 G.f.: (Sum_{k in Z} x^(3*k^2)) / (Sum_{k in Z} x^k^2).
%F A132002 G.f.: Product_{k>0} (1 + (-x)^k + x^(2*k)) / (1 - (-x)^k + x^(2*k)). - _Michael Somos_, Apr 04 2015
%F A132002 a(n) = (-1)^n * A098151(n).
%F A132002 Convolution inverse of A139137. Convolution square is A261320. - _Michael Somos_, Aug 14 2015
%F A132002 Expansion of f(-q, q^2) / f(q, -q^2) in powers of q where f(, ) is Ramanujan's general theta function. - _Michael Somos_, Nov 02 2015
%F A132002 a(n) = A139136(3*n) = A139137(3*n). - _Michael Somos_, Nov 02 2015
%F A132002 a(n) ~ (-1)^n * exp(Pi*sqrt(2*n/3)) / (2^(5/4) * 3^(3/4) * n^(3/4)). - _Vaclav Kotesovec_, Nov 16 2017
%e A132002 G.f. = 1 - 2*x + 4*x^2 - 6*x^3 + 10*x^4 - 16*x^5 + 24*x^6 - 36*x^7 + 52*x^8 + ...
%t A132002 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^3] / EllipticTheta[ 3, 0, q], {q, 0, n}]; (* _Michael Somos_, Apr 04 2015 *)
%t A132002 a[ n_] := SeriesCoefficient[ QPochhammer[ q, -q^3] QPochhammer[ -q^2, -q^3] / (QPochhammer[ -q, -q^3] QPochhammer[ q^2, -q^3]), {q, 0, n}]; (* _Michael Somos_, Nov 02 2015 *)
%o A132002 (PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=1, sqrtint(n\3), 2*x^(3*k^2), 1 + x * O(x^n)) / sum(k=1, sqrtint(n), 2*x^k^2, 1 + x * O(x^n)), n))};
%o A132002 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A)^5 / (eta(x^2 + A)^5 * eta(x^3 + A)^2 * eta(x^12 + A)^2), n))};
%Y A132002 Cf. A098151, A139007, A139136, A139137, A261320.
%K A132002 sign
%O A132002 0,2
%A A132002 _Michael Somos_, Aug 06 2007