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Expansion of f(x^5, -x^7) - x * f(-x, x^11) in powers of x where f() is Ramanujan's two-variable theta function.
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%I #15 Mar 12 2021 22:24:46

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

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

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

%N Expansion of f(x^5, -x^7) - x * f(-x, x^11) 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.

%H G. C. Greubel, <a href="/A206958/b206958.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^8) * f(-x^8,-x^8) / f(x,-x^3) in powers of x where f() is Ramanujan't two-variable theta function.

%F Euler transform of period 16 sequence [ -1, 1, 1, 0, 1, 0, -1, -2, -1, 0, 1, 0, 1, 1, -1, -1, ...].

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

%F G.f.: Product_{k>0} (1 - (-1)^k * x^(4*k-1)) * (1 + (-1)^k * x^(4*k-3)) * (1 - (-1)^k * x^(4*k)) * (1 + x^(8*k-6)) * (1 + x^(8*k-2)).

%F a(5*n + 3) = a(5*n + 4) = 0. |a(n)| = A080995(n).

%F a(n) = (-1)^n * A206959(n). - _Michael Somos_, Apr 01 2015

%e G.f. = 1 - x + x^2 + x^5 - x^7 - x^12 + x^15 - x^22 - x^26 + x^35 - x^40 - x^51 + ...

%e G.f. = q - q^25 + q^49 + q^121 - q^169 - q^289 + q^361 - q^529 - q^625 + q^841 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x^12] (QPochhammer[ -x^5, -x^12] QPochhammer[ x^7, -x^12] - x QPochhammer[ x, -x^12] QPochhammer[ -x^11, -x^12]), {x, 0, n}]; (* _Michael Somos_, Apr 01 2015 *)

%t a[ n_] := SeriesCoefficient[ Product[(1 - x^k)^{1, -1, -1, 0, -1, 0, 1, 2, 1, 0, -1, 0, -1, -1, 1, 1}[[Mod[k, 16, 1]]], {k, n}], {x, 0, n}]; (* _Michael Somos_, Apr 01 2015 *)

%o (PARI) {a(n) = my(m); if( issquare( 24*n + 1, &m), if( m%6 != 1, m = -m); m \= 6; (-1)^(m \ 4 + (m %4 == 2)), 0)};

%Y Cf. A080995, A206959.

%K sign

%O 0,1

%A _Michael Somos_, Feb 14 2012