A113431 - OEIS

%I #24 Mar 12 2021 22:24:43

%S 1,-1,-1,0,1,0,0,0,0,0,0,1,0,0,0,-1,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,

%T 0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,

%U 0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0

%N Expansion of f(-x) * f(-x^10) / f(-x^4, -x^6) in powers of x where f() is Ramanujan's two-variable theta function.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%C 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^3, b = x^2.

%C f(a,b) = Sum_{k} a^((k^2+k)/2)*b^((k^2-k)/2) is Ramanujan's two-variable theta function.

%H G. C. Greubel, <a href="/A113431/b113431.txt">Table of n, a(n) for n = 0..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/QuintupleProductIdentity.html">Quintuple Product Identity</a>

%F Expansion of f(x^4, x^11) - x * f(x, x^14) = f(-x^5, -x^10) * f(-x, -x^4) / f(x^2, x^3) in powers of x where f() is Ramanujan's two-variable theta function.

%F Expansion of H(x^2) * f(-x) where H() is g.f. of A003106.

%F Euler transform of period 10 sequence [ -1, -1, -1, 0, -1, 0, -1, -1, -1, -1, ...].

%F G.f.: Sum_{k} x^(5*k * (3*k + 1) / 2) * (x^(-6*k) - x^(6*k + 2)).

%F G.f.: Product_{k>0} (1 - x^k) / ((1 - x^(10*k - 4)) * (1 - x^(10*k - 6))).

%e 1 - x - x^2 + x^4 + x^11 - x^15 - x^18 + x^23 + x^37 - x^44 - x^49 + ...

%e q^49 - q^169 - q^289 + q^529 + q^1369 - q^1849 - q^2209 + q^2809 + q^4489 + ...

%t f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A113431[n_] := SeriesCoefficient[f[-x, -x^2]*f[-x^10, -x^20]/f[-x^4, -x^6], {x, 0, n}]; Table[A113431[n], {n,0,50}] (* _G. C. Greubel_, Jun 17 2017 *)

%o (PARI) {a(n) = local(m); if( n<0 || !issquare( n*120 + 49, &m) || 1! = gcd(m, 30),  0, (-1)^( m%30 \ 10))}

%o (PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, 1 - x^k*[ 1, 1, 1, 1, 0, 1, 0, 1, 1, 1][k%10 + 1], 1 + x * O(x^n)), n))}

%Y Cf. A003106.

%K sign

%O 0,1

%A _Michael Somos_, Oct 31 2005