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%I #29 Mar 12 2021 22:24:46
%S 1,-1,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,1,
%T 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,
%U 0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N a(n) = A010815(7*n + 5).
%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^4, b = -x^3.
%H G. C. Greubel, <a href="/A185175/b185175.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^5, -x^16) - x * f(-x^2, -x^19) = f(-x^7, -x^14) * f(-x, -x^6) / f(-x^3, -x^4) in powers of x where f(, ) is Ramanujan's general theta function.
%F Euler transform of period 7 sequence [-1, 0, 1, 1, 0, -1, -1, ...].
%F G.f.: Sum_{k in Z} (-1)^k * x^(7*k * (3*k + 1) / 2) * (x^(9*k + 3) + x^(-9*k)).
%F G.f.: Product_{k>0} (1 - x^(7*k)) * (1 - x^(7*k - 1)) * (1 - x^(7*k - 6)) / ((1 - x^(7*k - 3)) * (1 - x^(7*k - 4))).
%e G.f. = 1 - x + x^3 - x^5 - x^16 + x^20 - x^26 + x^31 + x^53 - x^60 + x^70 + ...
%e G.f. = q^121 - q^289 + q^625 - q^961 - q^2809 + q^3481 - q^4489 + q^5329 + q^9025 + ...
%t f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A185175[n_] := SeriesCoefficient[f[-x^7, -x^14]*f[-x, -x^6]/ f[-x^3, -x^4], {x, 0, n}]; Table[A185175[n], {n,0,50}] (* _G. C. Greubel_, Jun 19 2017 *)
%t nmax = 100; CoefficientList[Series[Product[(1 - x^(7*k)) * (1 - x^(7*k-1)) * (1 - x^(7*k-6)) / ((1 - x^(7*k-3)) * (1 - x^(7*k-4))), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 20 2017 *)
%t a[ n_] := With[{m = Sqrt[168 n + 121]}, If[ IntegerQ@m, KroneckerSymbol[ 12, m], 0]]; (* _Michael Somos_, Jun 27 2017 *)
%t a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{1, 0, -1, -1, 0, 1, 1}[[Mod[k, 7, 1]]], {k, n}], {x, 0, n}]; (* _Michael Somos_, Jun 27 2017 *)
%o (PARI) {a(n) = my(m); if( issquare( 168*n + 121, &m), kronecker( 12, m))};
%o (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x*O(x^n))^[1, 1, 0, -1, -1, 0, 1][k%7+1]), n))}; /* _Michael Somos_, Jun 27 2017 */
%Y Cf. A010815.
%K sign
%O 0,1
%A _Michael Somos_, Jan 21 2012