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A152243 Expansion of a(q) * f(-q)^4 where f() is a Ramanujan theta function and a() is a cubic AGM function.

%I A152243 #16 Feb 16 2025 08:33:09
%S A152243 1,2,-22,26,25,-46,26,-22,-45,0,74,122,-46,-142,0,-44,2,194,-214,0,
%T A152243 121,146,52,-22,0,-286,-118,-262,315,50,314,0,-382,386,0,-166,-92,338,
%U A152243 26,0,-286,-572,0,52,0,242,122,458,289,0,-44,-358,-142,0,-550,362,482,-188,-502,0,315,-718,698,-694
%N A152243 Expansion of a(q) * f(-q)^4 where f() is a Ramanujan theta function and a() is a cubic AGM function.
%C A152243 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A152243 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A152243 G. C. Greubel, <a href="/A152243/b152243.txt">Table of n, a(n) for n = 0..1000</a>
%H A152243 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A152243 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A152243 Expansion of q^(-1/2) * ( (eta(q) * eta(q^3))^3 + 3 * (eta(q^3) * eta(q^9))^3 ) in powers of q^3.
%F A152243 a(n) = b(6*n + 1) where b(n) is multiplicative with b(2^e) = b(3^e) = 0^e, 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 A152243 G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 15552^(1/2) (t / i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A152244.
%F A152243 a(n) = A030208(3*n) = A152244(3*n).
%e A152243 G.f. = 1 + 2*x - 22*x^2 + 26*x^3 + 25*x^4 - 46*x^5 + 26*x^6 - 22*x^7 + ...
%e A152243 G.f. = q + 2*q^7 - 22*q^13 + 26*q^19 + 25*q^25 - 46*q^31 + 26*q^37 - 22*q^43 + ...
%t A152243 eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(-1/6)*((eta[q^(1/3)]*eta[q^1])^3 + 3*(eta[q^1]*eta[q^3])^3), {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* _G. C. Greubel_, Jun 10 2018 *)
%o A152243 (PARI) {a(n) = local(A); if( n<0, 0, n *= 3; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A))^3 + 3 * x * (eta(x^3 + A) * eta(x^9 + A))^3, n))};
%Y A152243 Cf. A030208, A152244.
%K A152243 sign
%O A152243 0,2
%A A152243 _Michael Somos_, Nov 30 2008