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%I #17 Mar 12 2021 22:24:42
%S 1,1,0,1,-1,-1,1,-1,0,0,0,0,-1,-1,-1,0,1,1,-1,0,-1,0,0,-1,1,0,0,0,0,
%T -1,0,0,1,1,0,1,-1,-1,1,-1,0,1,0,1,-1,0,0,0,1,1,0,1,-1,-1,1,0,1,0,0,0,
%U -1,-1,0,-1,2,2,0,1,-1,-1,0,-1,1,1,0,0,-1,-1,0,0,2,1,-1,1,-2,-1,0,-1,1,1,-1,1,-1,-1,0,-1,2,2,0,1,-2,-2,1,-2,1,1,0,1,-2
%N Expansion of Product_{k>0} (1 + x^k) * (1 - x^(2*k)) / (1 + x^(4*k)) in powers of x.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C |a(n)| < 2 if n < 64.
%C A089385 and A000041 have the same parity. - _Vladeta Jovovic_, Dec 31 2003
%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 q^(1/24) * eta(q^2)^2 * eta(q^4) / (eta(q) * eta(q^8)) in powers of q. - _Michael Somos_, Sep 30 2011
%F Expansion of f(x) * chi(x^2) = psi(x) * chi(-x^4) = phi(-x^4) * chi(x) where phi(), psi(), chi(), f() are Ramanujan theta functions. - _Michael Somos_, Sep 30 2011
%F G.f.: Product_{k>0} (1 + x^k) * (1 - x^(2*k)) / (1 + x^(4*k)). - _Michael Somos_, Sep 30 2011
%F G.f.: Product_{k>=1} Sum_{n>=0} b(n)*x^(kn), where b(n)=-1 if n is congruent to 2, 3, 4, or 5 modulo 8, b(n)=+1 otherwise.
%F G.f.: Product_{k>0} (1-x^k)*(1+x^k)^2/(1+x^(4*k)). - _Vladeta Jovovic_, Dec 31 2003
%F Euler transform of period 8 sequence [1, -1, 1, -2, 1, -1, 1, -1, ...]. - _Vladeta Jovovic_, Aug 20 2004
%F A040051(n) == a(n) (mod 2).
%e 1 + x + x^3 - x^4 - x^5 + x^6 - x^7 - x^12 - x^13 - x^14 + x^16 + ...
%e 1/q + q^23 + q^71 - q^95 - q^119 + q^143 - q^167 - q^287 - q^311 + ...
%t a[ n_] := SeriesCoefficient[ Product[ (1 + x^k) (1 - x^(2 k)) / (1 + x^(4 k)), {k, n}], {x, 0, n}] (* _Michael Somos_, Sep 30 2011 *)
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ -x, x] / QPochhammer[ -x^4, x^4], {x, 0, n}] (* _Michael Somos_, Sep 30 2011 *)
%t a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, n}] Product[ 1 + x^k, {k, 2, n, 4}], {x, 0, n}] (* _Michael Somos_, Sep 30 2011 *)
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x] Product[ 1 + x^k, {k, 2, n, 4}], {x, 0, n}] (* _Michael Somos_, Sep 30 2011 *)
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ -x^2, x^4], {x, 0, n}] (* _Michael Somos_, Sep 30 2011 *)
%t a[ n_] := SeriesCoefficient[ Product[ 1 + x^k, {k, 1, n, 2}] EllipticTheta[ 3, 0, -x^4], {x, 0, n}] (* _Michael Somos_, Sep 30 2011 *)
%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] EllipticTheta[ 3, 0, -x^4], {x, 0, n}] (* _Michael Somos_, Sep 30 2011 *)
%t a[ n_] := SeriesCoefficient[ 1/2 Product[ 1 + (-x^4)^k, {k, 1, n/4, 2}] EllipticTheta[ 2, 0, x^(1/2)], {x, 0, n + 1/8}] (* _Michael Somos_, Sep 30 2011 *)
%t a[ n_] := SeriesCoefficient[ 1/2 QPochhammer[ x^4, x^8] EllipticTheta[ 2, 0, x^(1/2)], {x, 0, n + 1/8}] (* _Michael Somos_, Sep 30 2011 *)
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^4 + A) / (eta(x + A) * eta(x^8 + A)), n))} /* _Michael Somos_, Sep 30 2011 */
%Y Cf. A040051.
%K sign,easy
%O 0,65
%A _David S. Newman_, Dec 28 2003
%E More terms from _Vladeta Jovovic_, Dec 31 2003
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