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A133988 Expansion of phi(x) / chi(x^3) in powers of x where phi(), chi() are Ramanujan theta functions.

%I A133988 #23 Feb 16 2025 08:33:06
%S A133988 1,2,0,-1,0,0,1,0,0,0,-2,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,0,-2,0,0,0,
%T A133988 0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,
%U A133988 1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0
%N A133988 Expansion of phi(x) / chi(x^3) in powers of x where phi(), chi() are Ramanujan theta functions.
%C A133988 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A133988 This is an example of the quintuple product identity in the form f(a*b^4, a^2/b) - (a/b) * f(a^4*b, b^2/a) = f(-a*b, -a^2*b^2) * f(-a/b, -b^2) / f(a, b) where a = x^2, b = -x. - _Michael Somos_, Jan 21 2012
%C A133988 Number 13 of the 14 primitive eta-products which are holomorphic modular forms of weight 1/2 listed by D. Zagier on page 30 of "The 1-2-3 of Modular Forms". - _Michael Somos_, May 04 2016
%H A133988 Seiichi Manyama, <a href="/A133988/b133988.txt">Table of n, a(n) for n = 0..10000</a>
%H A133988 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A133988 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A133988 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuintupleProductIdentity.html">Quintuple Product Identity</a>
%F A133988 Expansion of q^(-1/8) * eta(q^2)^5 * eta(q^3) * eta(q^12) / ( eta(q) * eta(q^4) * eta(q^6) )^2 in powers of q.
%F A133988 Expansion of psi(-x) + 3 * x * psi(-x^9) in powers of x where psi() is a Ramanujan theta function.
%F A133988 Expansion of f(x, x^5) * f(x) / f(-x^6) = f(-x^3, x^6) + x * f(1, -x^9) in powers of x where f(, ) is Ramanujan's general theta function.
%F A133988 Expansion of phi(x^9) / chi(x^3) + 2 * x * psi(-x^9) in powers of x where phi(), chi() are Ramanujan theta functions.
%F A133988 Euler transform of period 12 sequence [ 2, -3, 1, -1, 2, -2, 2, -1, 1, -3, 2, -1, ...].
%F A133988 a(n) = b(8*n + 1) where b() is multiplicative with b(3^(2*e)) = -2 * (-1)^e, b(p^(2*e)) = (-1)^e if p == 3, 5 (mod 8), b(p^(2*e)) = +1 if p == 1, 7 (mod 8) and b(p^(2*e-1)) = b(2^e) = 0 if e>0.
%F A133988 G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = 12 (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A133985.
%F A133988 G.f.: Sum_{k>0} (-1)^floor(k/2) * (x^((k^2 - k)/2) + 3 * x^(9*(k^2 - k)/2 + 1) ).
%F A133988 G.f.: Sum_{k in Z} (-1)^(k + floor(k/2)) * x^(3*k * (3*k + 1) / 2) * ( x^(-3*k) + x^(3*k + 1) ).
%F A133988 a(n) = (-1)^n * A089812(n).
%F A133988 a(3*n) = A133985(n). a(3*n + 2) = 0. - _Michael Somos_, Oct 30 2015
%e A133988 G.f. = 1 + 2*x - x^3 + x^6 - 2*x^10 - x^15 - x^21 - 2*x^28 + x^36 - x^45 + ...
%e A133988 G.f. = q + 2*q^9 - q^25 + q^49 - 2*q^81 - q^121 - q^169 - 2*q^225 + q^289 - ...
%t A133988 a[ n_] := (-1)^Quotient[n, 3] Mod[n + 1, 3] Boole[ IntegerQ[ Sqrt[8 n + 1]]]; (* _Michael Somos_, Oct 30 2015 *)
%t A133988 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x]  QPochhammer[ x^3, -x^3], {x, 0, n}]; (* _Michael Somos_, Oct 30 2015 *)
%o A133988 (PARI) {a(n) = (-1)^(n\3) * ((n + 1)%3) * issquare( 8*n + 1)};
%o A133988 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A))^2, n))};
%Y A133988 Cf. A089812, A133985.
%K A133988 sign
%O A133988 0,2
%A A133988 _Michael Somos_, Oct 01 2007