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A214529 Expansion of f(x^29, -x^31) - x * f(x^19, -x^41) + x^3 * f(x^11, -x^49) - x^7 * f(-x, x^59) in powers of x where f() is Ramanujan's two-variable theta function.

%I A214529 #15 Feb 16 2025 08:33:18
%S A214529 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 A214529 0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A214529 -1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A214529 Expansion of f(x^29, -x^31) - x * f(x^19, -x^41) + x^3 * f(x^11, -x^49) - x^7 * f(-x, x^59) in powers of x where f() is Ramanujan's two-variable theta function.
%C A214529 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A214529 G. C. Greubel, <a href="/A214529/b214529.txt">Table of n, a(n) for n = 0..1000</a>
%H A214529 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A214529 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A214529 |a(n)| is the characteristic function of A093722.
%F A214529 The exponents in the q-series q * A(q^120) are the squares of the numbers in A057538.
%F A214529 Euler transform of a period 80 sequence.
%F A214529 G.f.: Sum_{k} (-1)^(floor((k - 1)/2) + floor(k/4)) * x^(3*k * (5*k + 1)/2) * (x^(4*k + 1) + x^(-16*k + 7)).
%F A214529 a(n) = (-1)^n * A208546(n).
%e A214529 1 - x + x^3 - x^7 + x^8 + x^14 - x^20 + x^29 - x^31 + x^42 - x^52 - x^66 + ...
%e A214529 q - q^121 + q^361 - q^841 + q^961 + q^1681 - q^2401 + q^3481 - q^3721 + ...
%t A214529 a[ n_] := Module[ {m}, If[ n >= 0 && OddQ[ DivisorSigma[ 0, 120 n + 1]], m = Sqrt[ 120 n + 1]; (-1)^(Quotient[ m, 40] + Quotient[ m, 3]), 0]]; Table[a[n], {n, 0, 30}]
%o A214529 (PARI) {a(n) = local(m); if( issquare( 120*n + 1, &m), (-1)^(m \ 40 + m \ 3))}
%Y A214529 Cf. A057538, A093722, A208546.
%K A214529 sign
%O A214529 0,1
%A A214529 _Michael Somos_, Jul 20 2012