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A204850 Expansion of f(x)^3 - 9 * x * f(x^9)^3 in powers of x where f() is a Ramanujan theta function.

%I A204850 #17 Feb 16 2025 08:33:16
%S A204850 1,-6,0,-5,0,0,-7,0,0,0,-18,0,0,0,0,11,0,0,0,0,0,-13,0,0,0,0,0,0,30,0,
%T A204850 0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,19,0,0,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,
%U A204850 0,0,0,-23,0,0,0,0,0,0,0,0,0,0,0,25,0,0,0
%N A204850 Expansion of f(x)^3 - 9 * x * f(x^9)^3 in powers of x where f() is a Ramanujan theta function.
%C A204850 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A204850 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%H A204850 G. C. Greubel, <a href="/A204850/b204850.txt">Table of n, a(n) for n = 0..1000</a>
%H A204850 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A204850 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A204850 Expansion of f(x^3) * a(-x) in powers of x where f() is a Ramanujan theta function and a() is a cubic AGM theta function.
%F A204850 a(n) = b(8*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = - (1 + (-1)^e) * 3^(e/2), b(p^e) = (1 + (-1)^e)/2 * p^(e/2) if p == 1, 3 (mod 8), b(p^e) = (1 + (-1)^e)/2 * (-p)^(e/2) if p == 5, 7 (mod 8). - _Michael Somos_, Jun 19 2015
%F A204850 G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = -576 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A133079.
%F A204850 a(3*n + 2) = a(5*n + 2) = a(5*n + 4) = a(9*n + 4) = a(9*n + 7) = 0. a(3*n) = A133079(n). a(9*n + 1) = -6 * A133089(n). a(25*n + 3) = -5 * a(n). a(n) = (-1)^n * A202394(n).
%e A204850 G.f. = 1 - 6*x - 5*x^3 - 7*x^6 - 18*x^10 + 11*x^15 - 13*x^21 + 30*x^28 + ...
%e A204850 G.f. = q - 6*q^9 - 5*q^25 - 7*q^49 - 18*q^81 + 11*q^121 - 13*q^169 + ...
%t A204850 a[ n_] := With[ {m = Sqrt[8 n + 1]}, If[ IntegerQ@m, m (-1)^(n + Quotient[m, 6]), 0] If[ Divisible[ m, 3], 2, 1]]; (* _Michael Somos_, Jun 19 2015 *)
%t A204850 a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^3 - 9 x QPochhammer[ -x^9]^3, {x, 0, n}]; (* _Michael Somos_, Jun 19 2015 *)
%o A204850 (PARI) {a(n) = my(m); if( issquare(8*n + 1, &m), (-1)^(m \ 6 + n) * m * ((m%3 == 0) + 1), 0)};
%o A204850 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(-x + A)^3 - 9 * x * eta(-x^9 + A)^3, n))};
%Y A204850 Cf. A133079, A133089, A202394.
%K A204850 sign
%O A204850 0,2
%A A204850 _Michael Somos_, Jan 19 2012