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%I #18 Mar 12 2021 22:24:46
%S 1,1,0,0,0,-1,0,0,0,0,-1,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,1,0,
%T 0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,
%U 0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,-1,0,0,0,0,0,0,0,0
%N a(n) = A010815(7*n).
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C 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^6, b = -x.
%H G. C. Greubel, <a href="/A185059/b185059.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>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/QuintupleProductIdentity.html">Quintuple Product Identity</a>
%F Expansion of f(-x^10, -x^11) + x * f(-x^4, -x^17) = f(-x^7, -x^14) * f(-x^2, -x^5) / f(-x, -x^6) in powers of x where f() is Ramanujan's two-variable theta function.
%F Euler transform of period 7 sequence [ 1, -1, 0, 0, -1, 1, -1, ...].
%F Sum_{k} (-1)^k * x^(7*k * (3*k + 1) / 2) * (x^(3*k + 1) + x^(-3*k)).
%F Product_{k>0} (1 - x^(7*k)) * (1 - x^(7*k - 2)) * (1 - x^(7*k - 5)) / ((1 - x^(7*k - 1)) * (1 - x^(7*k - 6))).
%e 1 + x - x^5 - x^10 - x^11 - x^18 + x^30 + x^41 + x^43 + x^56 - x^76 + ...
%e q + q^169 - q^841 - q^1681 - q^1849 - q^3025 + q^5041 + q^6889 + q^7225 + ...
%t f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A185059[n_] := SeriesCoefficient[f[-x^7, -x^14]*f[-x^2, -x^5]/ f[-x, -x^6], {x, 0, n}]; Table[A185059[n], {n,0,50}] (* _G. C. Greubel_, Jun 19 2017 *)
%t nmax = 100; CoefficientList[Series[Product[(1 - x^(7*k)) * (1 - x^(7*k-2)) * (1 - x^(7*k-5)) / ((1 - x^(7*k-1)) * (1 - x^(7*k-6))), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 20 2017 *)
%o (PARI) {a(n) = local(m); if( issquare( 168*n + 1, &m), kronecker( 12, m))}
%Y Cf. A010815.
%K sign
%O 0,1
%A _Michael Somos_, Jan 21 2012
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