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A133079 Expansion of f(x)^3 - 3 * x * f(x^9)^3 in powers of x^3 where f() is a Ramanujan theta function.

%I A133079 #26 Feb 16 2025 08:33:06
%S A133079 1,-5,-7,0,0,11,0,-13,0,0,0,0,17,0,0,19,0,0,0,0,0,0,-23,0,0,0,25,0,0,
%T A133079 0,0,0,0,0,0,-29,0,0,0,0,-31,0,0,0,0,0,0,0,0,0,0,35,0,0,0,0,0,-37,0,0,
%U A133079 0,0,0,0,0,0,0,0,0,0,41,0,0,0,0,0,0,43,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-47,0,0,0
%N A133079 Expansion of f(x)^3 - 3 * x * f(x^9)^3 in powers of x^3 where f() is a Ramanujan theta function.
%C A133079 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A133079 Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%C A133079 There is a plus sign on the left side and the first and third plus signs on the right side which should be minuses in Ramanujan's equation.
%D A133079 B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 357, Entry 5, Eq. (5.1)
%D A133079 S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 1, see page 266.
%H A133079 Seiichi Manyama, <a href="/A133079/b133079.txt">Table of n, a(n) for n = 0..10000</a>
%H A133079 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A133079 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A133079 Expansion of f(x) * a(-x) in powers of x where f() is a Ramanujan theta function and a() is a cubic AGM theta function.
%F A133079 G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = -192 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A204850.
%F A133079 a(n) = b(24*n + 1) where b(n) is multiplicative with b(2^e) = b(3^e) = 0^e, 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).
%F A133079 G.f.: Sum_{k in Z} Kronecker( 2, 2*k + 1) * (6*k + 1) * x^(k * (3*k + 1)/2).
%F A133079 a(5*n + 3) = a(5*n + 4) = 0. a(25*n + 1) = -5 * a(n). a(n) = (-1)^n * A116916(n).
%F A133079 a(n) = A133089(3*n) = A204850(3*n). - _Michael Somos_, Jun 19 2015
%e A133079 G.f. = 1 - 5*x - 7*x^2 + 11*x^5 - 13*x^7 + 17*x^12 + 19*x^15 - 23*x^22 + ...
%e A133079 G.f. = q - 5*q^25 - 7*q^49 + 11*q^121 - 13*q^169 + 17*q^289 + 19*q^361 - ...
%t A133079 a[ n_] := With[ {m = Sqrt[24 n + 1]}, If[ IntegerQ@m, m (-1)^Boole[Mod[m, 8] > 4], 0]]; (* _Michael Somos_, Jun 19 2015 *)
%t A133079 a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^3 - 3 x QPochhammer[ -x^9]^3, {x, 0, 3 n}]; (* _Michael Somos_, Jun 19 2015 *)
%o A133079 (PARI) {a(n) = if( issquare( 24*n + 1, &n), n * (-1) ^ (n%8 > 4), 0)};
%o A133079 (PARI) {a(n) = my(A, p, e); if( n<0, 0, n = 24*n + 1; A = factor(n); prod(k = 1, matsize(A) [1], [p, e] = A[k,]; if( p < 5, 0, p *= kronecker( -2, p); if( e%2, 0, p^(e/2) ))))};
%o A133079 (PARI) {a(n) = my(A); if( n<0, 0, n *= 3; A = x * O(x^n); polcoeff( eta(-x + A)^3 - 3 * x * eta(-x^9 + A)^3, n))};
%Y A133079 Cf. A116916, A133089, A204850.
%K A133079 sign
%O A133079 0,2
%A A133079 _Michael Somos_, Sep 08 2007