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A218171 Expansion of f(x^11, x^13) - x * f(x^5, x^19) in powers of x where f(, ) is Ramanujan's general theta function.

%I A218171 #32 Aug 31 2025 10:45:59
%S A218171 1,-1,0,0,0,0,-1,0,0,0,0,1,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,
%T A218171 0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,
%U A218171 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A218171 Expansion of f(x^11, x^13) - x * f(x^5, x^19) in powers of x where f(, ) is Ramanujan's general theta function.
%C A218171 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A218171 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^7, b = x. - _Michael Somos_, Nov 09 2014
%C A218171 This is exercise 0.13 3 on page 45 of Cooper [2017]. - _Michael Somos_, Aug 30 2025
%H A218171 G. C. Greubel, <a href="/A218171/b218171.txt">Table of n, a(n) for n = 0..10000</a>
%H A218171 Shaun Cooper, <a href="https://doi.org/10.1007/978-3-319-56172-1">Ramanujan's Theta Functions</a>, Springer International (2017).
%H A218171 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A218171 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A218171 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuintupleProductIdentity.html">Quintuple Product Identity</a>
%F A218171 Expansion of f(x^3, x^5) * chi(-x) in powers of x where f(, ) is Ramanujan's general theta function and chi() is a Ramanujan theta function.
%F A218171 G.f.: Sum_{k in Z} x^(12*k^2 + k) - x^(12*k^2 + 7*k + 1).
%F A218171 a(n) = A010815(2*n) for all n in Z.
%F A218171 G.f.: Product_{j>0} (1-x^(8*j-1)) * (1-x^(8*j-7)) * (1-x^(8*j)) * (1-x^(16*j-6)) * (1-x^(16*j-10)). [Cooper 2017] - _Michael Somos_, Aug 30 2025
%e A218171 G.f. = 1 - x - x^6 + x^11 + x^13 - x^20 - x^35 + x^46 + x^50 - x^63 - x^88 + ...
%e A218171 G.f. = q - q^49 - q^289 + q^529 + q^625 - q^961 - q^1681 + q^2209 + q^2401 + ...
%t A218171 a[ n_] := If[ n < 0, 0, If[ OddQ[ DivisorSigma[ 0, 48 n + 1]], JacobiSymbol[ 6, Sqrt[48 n + 1]], 0]]; (* _Michael Somos_, Nov 09 2014 *)
%t A218171 a[ n_] := SeriesCoefficient[ (QPochhammer[ -q] + QPochhammer[ q]) / 2, {q, 0, 2 n}]; (* _Michael Somos_, Nov 09 2014 *)
%t A218171 a[ n_] := SeriesCoefficient[ QPochhammer[ q] (QPochhammer[ q^2]^3 / QPochhammer[ q]^2/ QPochhammer[ q^4] + 1) / 2, {q, 0, 2 n}]; (* _Michael Somos_, Nov 09 2014 *)
%o A218171 (PARI) {a(n) = my(m); if( issquare(48*n + 1, &m), kronecker(6, m), 0)};
%o A218171 (PARI) {a(n) = my(m); if( n<0, 0, m = 2*n; polcoeff( eta(x + x * O(x^m)), m))};
%Y A218171 Cf. A010815, A115671.
%K A218171 sign,changed
%O A218171 0,1
%A A218171 _Michael Somos_, Oct 22 2012