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A116915 Expansion of f(-x, -x^4)^2 / f(-x, -x^2) in powers of x where f(, ) is Ramanujan's general theta function.

%I A116915 #16 Feb 16 2025 08:33:00
%S A116915 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 A116915 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 A116915 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 A116915 Expansion of f(-x, -x^4)^2 / f(-x, -x^2) in powers of x where f(, ) is Ramanujan's general theta function.
%C A116915 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A116915 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. - _Michael Somos_, Jul 12 2012
%H A116915 G. C. Greubel, <a href="/A116915/b116915.txt">Table of n, a(n) for n = 0..1000</a>
%H A116915 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A116915 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A116915 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuintupleProductIdentity.html">Quintuple Product Identity</a>
%F A116915 Expansion of f(-x^4, -x^11) - x * f(-x, -x^14) where f() is Ramanujan's two-variable theta function. - _Michael Somos_, Nov 08 2015
%F A116915 Euler transform of period 5 sequence [ -1, 1, 1, -1, -1, ...].
%F A116915 G.f.: Sum_{k in Z} (-1)^k * (x^((15*k^2 - 7*k)/2) - x^((15*k^2 + 13*k)/2 + 1)).
%F A116915 G.f.: Product_{k>0} (1 - x^(5*k)) * (1 - x^(5*k - 1)) * (1 - x^(5*k - 4)) / ((1 - x^(5*k - 2)) * (1 - x^(5*k - 3))).
%F A116915 - a(n) = A010815(5*n + 2).
%e A116915 G.f. = 1 - x + x^2 - x^4 - x^11 + x^15 - x^18 + x^23 + x^37 - x^44 + x^49 - x^57 + ...
%e A116915 G.f. = q^49 - q^169 + q^289 - q^529 - q^1369 + q^1849 - q^2209 + q^2809 + q^4489 + ...
%t A116915 a[ n_] := SeriesCoefficient[ (QPochhammer[ x^5] QPochhammer[ x, x^5] QPochhammer[ x^4, x^5])^2 / QPochhammer[ x], {x, 0, n}]; (* _Michael Somos_, Jul 12 2012 *)
%t A116915 a[ n_] := With[ {k = Sqrt[ 120 n + 49]}, If[ IntegerQ[ k], -KroneckerSymbol[ 12, k], 0]]; (* _Michael Somos_, Nov 08 2015 *)
%o A116915 (PARI) {a(n) = n = 5*n + 2; if( issquare(24*n + 1, &n), -kronecker( 12, n))};
%o A116915 (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))};
%Y A116915 Cf. A010815.
%K A116915 sign
%O A116915 0,1
%A A116915 _Michael Somos_, Feb 26 2006