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A232714 Expansion of f(-x, -x^6) in powers of x where f is Ramanujan's two-variable theta function.

%I A232714 #20 Jun 14 2025 07:16:38
%S A232714 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 A232714 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 A232714 0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0
%N A232714 Expansion of f(-x, -x^6) in powers of x where f is Ramanujan's two-variable theta function.
%C A232714 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A232714 Seiichi Manyama, <a href="/A232714/b232714.txt">Table of n, a(n) for n = 0..10000</a>
%H A232714 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A232714 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A232714 Euler transform of period 7 sequence [ -1, 0, 0, 0, 0, -1, -1, ...].
%F A232714 G.f.: Sum_{k in Z} (-1)^k * x^(k * (7*k + 5) / 2).
%F A232714 G.f.: Product_{k>0} (1 - x^(7*k-6)) * (1 - x^(7*k-1)) * (1 - x^(7*k)).
%F A232714 a(3*n + 2) = a(5*n + 2) = a(5*n + 3) = 0.
%F A232714 Convolution inverse of A195849.
%F A232714 abs(a(n)) = A274179(n). - _Michael Somos_, Jan 28 2017
%F A232714 a(n) = -(1/n)*Sum_{k=1..n} A284363(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, Mar 25 2017
%e A232714 G.f. = 1 - x - x^6 + x^9 + x^19 - x^24 - x^39 + x^46 + x^66 - x^75 - x^100 + ...
%e A232714 G.f. = q^25 - q^81 - q^361 + q^529 + q^1089 - q^1369 - q^2209 + q^2601 + q^3721 + ...
%t A232714 a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^7] QPochhammer[ q^6, q^7] QPochhammer[ q^7], {q, 0, n}];
%o A232714 (PARI) {a(n) = my(m); if( issquare( 56*n + 25, &m), (-1)^round( m / 14), 0)};
%Y A232714 Cf. A195849, A274179.
%K A232714 sign
%O A232714 0,1
%A A232714 _Michael Somos_, Nov 28 2013