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A132302 Expansion of f(-x, -x^5) * f(-x^6) / f(-x)^2 in powers of x where f(, ) and f() are Ramanujan theta functions.

%I A132302 #14 Feb 16 2025 08:33:06
%S A132302 1,1,3,5,10,15,26,39,63,92,140,201,295,415,591,818,1140,1554,2126,
%T A132302 2861,3855,5126,6816,8970,11793,15372,20007,25857,33356,42771,54734,
%U A132302 69683,88530,111968,141312,177642,222842,278557,347484,432095,536230,663549,819504
%N A132302 Expansion of f(-x, -x^5) * f(-x^6) / f(-x)^2 in powers of x where f(, ) and f() are Ramanujan theta functions.
%C A132302 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A132302 G. C. Greubel, <a href="/A132302/b132302.txt">Table of n, a(n) for n = 0..1000</a>
%H A132302 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A132302 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A132302 Expansion of q^(-1/2) * eta(q^6)^3 / (eta(q) * eta(q^2) * eta(q^3)) in powers of q.
%F A132302 Euler transform of period 6 sequence [ 1, 2, 2, 2, 1, 0, ...].
%F A132302 Given g.f. A(x), then B(q) = A(q^2) * q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u^2 - v)^3 - 4 * v^4 * (v - 3*u^2) * (2*v - 3*u^2).
%F A132302 G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (1/6) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132301.
%F A132302 a(n) = A124243(2*n + 1) = A132180(2*n + 1) = A132975(2*n + 1) = A213267(2*n + 1). - _Michael Somos_, Nov 01 2015
%F A132302 a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (2^(7/4)*3^(3/2)*n^(3/4)). - _Vaclav Kotesovec_, Nov 16 2017
%e A132302 G.f. = 1 + x + 3*x^2 + 5*x^3 + 10*x^4 + 15*x^5 + 26*x^6 + 39*x^7 + ...
%e A132302 G.f. = q + q^3 + 3*q^5 + 5*q^7 + 10*q^9 + 15*q^11 + 26*q^13 + 39*q^15 + ...
%t A132302 a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^6] QPochhammer[ x^5, x^6] QPochhammer[ x^6]^2 / QPochhammer[ x]^2, {x, 0, n}]; (* _Michael Somos_, Nov 01 2015 *)
%o A132302 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A)^3 / (eta(x + A) * eta(x^2 + A) * eta(x^3 + A)), n))};
%Y A132302 Cf. A124243, A132180, A132301, A132975, A213267.
%K A132302 nonn
%O A132302 0,3
%A A132302 _Michael Somos_, Aug 17 2007