%I #15 Mar 12 2021 22:24:47

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

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

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

%N Expansion of f(-x, -x^6) 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).

%H Seiichi Manyama, <a href="/A232714/b232714.txt">Table of n, a(n) for n = 0..10000</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 Euler transform of period 7 sequence [ -1, 0, 0, 0, 0, -1, -1, ...].

%F G.f.: Sum_{k in Z} (-1)^k * x^(k * (7*k + 5) / 2).

%F G.f.: Product_{k>0} (1 - x^(7*k-6)) * (1 - x^(7*k-1)) * (1 - x^(7*k)).

%F a(3*n + 2) = a(5*n + 2) = a(5*n + 3) = 0.

%F Convolution inverse of A195849.

%F abs(a(n)) = A274179(n). - _Michael Somos_, Jan 28 2017

%F a(n) = -(1/n)*Sum_{k=1..n} A284363(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, Mar 25 2017

%e G.f. = 1 - x - x^6 + x^9 + x^19 - x^24 - x^39 + x^46 + x^66 - x^75 - x^100 + ...

%e G.f. = q^25 - q^81 - q^361 + q^529 + q^1089 - q^1369 - q^2209 + q^2601 + q^3721 + ...

%t a[ n_] := SeriesCoefficient[ SeriesCoefficient[ QPochhammer[ q, q^7] QPochhammer[ q^6, q^7] QPochhammer[ q^7], {q, 0, n}];

%o (PARI) {a(n) = my(m); if( issquare( 56*n + 25, &m), (-1)^round( m / 14), 0)};

%Y Cf. A195849, A274179.

%K sign

%O 0,1

%A _Michael Somos_, Nov 28 2013