%I #17 Mar 12 2021 22:24:46
%S 1,1,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,
%T 0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,
%U 0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0
%N Expansion of f(-x^2, -x^3) * f(-x^2, -x^4) / f(-x, -x^2) in powers of x where f(, ) is Ramanujan's general theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A204220/b204220.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>
%F Expansion of G(x) * f(-x^2) where G() is the g.f. of A003114.
%F Expansion of f(-x^13, -x^17) + x * f(-x^7, -x^23) in powers of x.
%F Euler transform of period 10 sequence [ 1, -1, 0, 0, 0, 0, 0, -1, 1, -1, ...].
%F G.f.: Sum_{k} (-1)^k * x^(15*k^2) * (x^(2*k) + x^(8*k + 1)) = Product_{k>0} (1 - x^(10*k)) * (1 - x^(10*k -2)) * (1 - x^(10*k -8)) / ((1 - x^(10*k -1)) * (1 - x^(10*k -9))).
%F |a(n)| is the characteristic function of A204221.
%F The exponents in the q-series q * A(q^15) are the squares of the numbers == +- 1 or +- 4 (mod 15).
%e G.f. = 1 + x - x^8 - x^13 - x^17 - x^24 + x^45 + x^56 + x^64 + x^77 - x^112 + ...
%e G.f. = q + q^16 - q^121 - q^196 - q^256 - q^361 + q^676 + q^841 + q^961 + q^1156 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, x^5] QPochhammer[ x^3, x^5] QPochhammer[ x^5] QPochhammer[ -x, x], {x, 0, n}]; (* _Michael Somos_, Jan 06 2016 *)
%t a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{ -1, 1, 0, 0, 0, 0, 0, 1, -1, 1}[[Mod[k, 10, 1]]], {k, n}], {x, 0, n}]; (* _Michael Somos_, Jan 06 2016 *)
%o (PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k) ^ ([1, -1, 1, 0, 0, 0, 0, 0, 1, -1][k%10 + 1]), 1 + x * O(x^n)), n))};
%Y Cf. A003114, A010815.
%K sign
%O 0,1
%A _Michael Somos_, Jan 13 2012