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A113430 Expansion of f(-x, -x^2) * f(-x^10, -x^20) / f(-x^2, -x^8) in powers of x where f(, ) is Ramanujan's general theta function.

%I A113430 #26 Feb 16 2025 08:32:59
%S A113430 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 A113430 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,
%U A113430 0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A113430 Expansion of f(-x, -x^2) * f(-x^10, -x^20) / f(-x^2, -x^8) in powers of x where f(, ) is Ramanujan's general theta function.
%C A113430 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A113430 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.
%D A113430 George E. Andrews, Richard Askey and Ranjan Roy, Special Functions, Cambridge University Press, 1999.
%H A113430 G. C. Greubel, <a href="/A113430/b113430.txt">Table of n, a(n) for n = 0..1000</a>
%H A113430 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A113430 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A113430 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuintupleProductIdentity.html">Quintuple Product Identity</a>
%F A113430 Expansion of f(x^7, x^8) - x * f(x^2, x^13) in power of x.
%F A113430 Expansion of G(x^2) * f(-x) where G() is the g.f. of A003114.
%F A113430 Euler transform of period 10 sequence [ -1, 0, -1, -1, -1, -1, -1, 0, -1, -1, ...].
%F A113430 |a(n)| is the characteristic function of the numbers in A093722.
%F A113430 The exponents in the q-series q * A(q^120) are the square of the numbers in A057538.
%F A113430 G.f.: Prod_{k>0} (1 - x^k) / ((1 - x^(10*k - 2)) * (1 - x^(10*k - 8))) = Sum_{k in Z} x^((15*k^2 + k) / 2) - x^((15*k^2 - 11*k + 2) / 2).
%F A113430 A(q^2) = 1 + Sum_{n >= 0} q^(n^2) * Product_{k >= 2*n+1} 1 - q^k = 1 - q^2 - q^6 + q^14 + q^16 - q^28 - q^40  +  + - - . See Andrews et al., p. 591, Exercise 6(a). - _Peter Bala_, Dec 22 2024
%e A113430 G.f. = 1 - x - x^3 + x^7 + x^8 - x^14 - x^20 + x^29 + x^31 - x^42 - x^52 + ...
%e A113430 G.f. = q - q^121 - q^361 + q^841 + q^961 - q^1681 - q^2401 + q^3481 + q^3721 + ...
%t A113430 a[ n_] := SeriesCoefficient[ QPochhammer[ x] / (QPochhammer[ x^2, x^10] QPochhammer[ x^8, x^10]), {x, 0, n}]; (* _Michael Somos_, Jan 06 2016 *)
%o A113430 (PARI) {a(n) = my(m); if( n<0 || !issquare( n*120 + 1, &m) || 1!=gcd(m, 30), 0, (-1)^(m%30\10))};
%o A113430 (PARI)  {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, 1 - x^k * [1, 1, 0, 1, 1, 1, 1, 1, 0, 1][k%10 + 1], 1 + x * O(x^n)), n))};
%Y A113430 Cf. A003114, A010815, A057538, A093722.
%K A113430 sign,easy
%O A113430 0,1
%A A113430 _Michael Somos_, Oct 31 2005