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A152244 Expansion of a(x) * f(-x^3)^4 in powers of x where f() is a Ramanujan theta function and a() is a cubic AGM function.

%I A152244 #13 Feb 16 2025 08:33:09
%S A152244 1,6,0,2,-18,0,-22,0,0,26,12,0,25,54,0,-46,0,0,26,-132,0,-22,0,0,-45,
%T A152244 0,0,0,156,0,74,-36,0,122,0,0,-46,150,0,-142,-162,0,0,0,0,-44,-276,0,
%U A152244 2,0,0,194,0,0,-214,156,0,0,396,0,121,0,0,146,-132,0,52,0,0,-22,0,0,0,-270,0,-286,0,0,-118
%N A152244 Expansion of a(x) * f(-x^3)^4 in powers of x where f() is a Ramanujan theta function and a() is a cubic AGM function.
%C A152244 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A152244 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A152244 G. C. Greubel, <a href="/A152244/b152244.txt">Table of n, a(n) for n = 0..1000</a>
%H A152244 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A152244 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A152244 Expansion of q^(-1/2) *( (eta(q)*eta(q^3))^3 + 9*(eta(q^3)*eta(q^9))^3 ) in powers of q.
%F A152244 a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = -2 * (-3)^e if e>0, b(p^e) = p^e * (1 + (-1)^e) / 2 if p == 5 (mod 6), b(p^e) = b(p) * b(p^(e-1)) - p^2 * b(p^(e-2)) if p == 1 (mod 6).
%F A152244 G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 139968^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A152243.
%F A152244 a(3*n) = A152243(n). a(3*n + 1) = 6 * A030208(n). a(3*n + 2) = 0.
%e A152244 G.f. = 1 + 6*x + 2*x^3 - 18*x^4 - 22*x^6 + 26*x^9 + 12*x^10 + 25*x^11 + ...
%e A152244 G.f. = q + 6*q^3 + 2*q^7 - 18*q^9 - 22*q^13 + 26*q^19 + 12*q^21 + 25*q^25 + ...
%t A152244 a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^3])^3 + 9 x (QPochhammer[ x^3] QPochhammer[ x^9])^3 , {x, 0, n}]; (* _Michael Somos_, Sep 02 2015 *)
%o A152244 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A))^3 + 9 * x * (eta(x^3 + A) * eta(x^9 + A))^3, n))};
%Y A152244 Cf. A030208, A152243.
%K A152244 sign
%O A152244 0,2
%A A152244 _Michael Somos_, Nov 30 2008