cp's OEIS Frontend

A113681 Expansion of f(-x^2, -x^3)^2 / f(-x, -x^2) in powers of x where f() is Ramanujan's two-variable theta function.

%I A113681 #24 Feb 16 2025 08:32:59
%S A113681 1,1,0,-1,0,0,0,-1,-1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,1,
%T A113681 0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A113681 -1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0
%N A113681 Expansion of f(-x^2, -x^3)^2 / f(-x, -x^2) in powers of x where f() is Ramanujan's two-variable theta function.
%C A113681 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A113681 f(a,b) = Sum_{k} a^((k^2+k)/2)*b^((k^2-k)/2) is Ramanujan's two-variable theta function.
%C A113681 |a(n)| is the characteristic function of A093722.
%C A113681 The exponents in the q-series for this sequence are the squares of the numbers of A057538.
%C A113681 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^4, b = -x.
%H A113681 G. C. Greubel, <a href="/A113681/b113681.txt">Table of n, a(n) for n = 0..1000</a>
%H A113681 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A113681 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuintupleProductIdentity.html">Quintuple Product Identity</a>
%F A113681 Expansion of f(-x^7, -x^8) + x * f(-x^2, -x^13) where f() is Ramanujan's two-variable theta function.
%F A113681 Euler transform of period 5 sequence [ 1, -1, -1, 1, -1, ...].
%F A113681 G.f.: Sum_{k} (-1)^k * x^(5*k * (3*k + 1)/2) * (x^(-3*k) + x^(3*k + 1)).
%F A113681 G.f.: Product_{k>0} (1 - x^(5*k)) * (1 - x^(5*k - 2)) * (1 - x^(5*k - 3)) / ((1 - x^(5*k - 1)) * (1 - x^(5*k - 4))).
%F A113681 A010815(5*n) = a(n).
%e A113681 1 + x - x^3 - x^7 - x^8 - x^14 + x^20 + x^29 + x^31 + x^42 - x^52 - x^66 - ...
%e A113681 q + q^121 - q^361 - q^841 - q^961 - q^1681 + q^2401 + q^3481 + q^3721 + ...
%t A113681 a[ n_] := SeriesCoefficient[ (QPochhammer[ q^5] QPochhammer[ q^2, q^5] QPochhammer[ q^3, q^5])^2 / QPochhammer[ q], {q, 0, n}] (* _Michael Somos_, Jul 17 2012 *)
%o A113681 (PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k)^((k%5==0) - kronecker( 5, k)), 1 + x * O(x^n)), n))}
%o A113681 (PARI) {a(n) = n*=5; if( issquare( 24*n + 1, &n), kronecker( 12, n))}
%Y A113681 Cf. A010815, A057538, A093722, A113430.
%K A113681 sign
%O A113681 0,1
%A A113681 _Michael Somos_, Nov 04 2005