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%I #26 Mar 12 2021 22:24:45
%S 1,1,-2,-1,0,-2,1,0,0,2,1,2,-2,0,2,1,0,-2,0,-2,0,-1,0,0,-2,0,0,0,-1,2,
%T -2,0,2,0,0,2,3,0,0,-2,0,0,2,0,2,1,-2,0,0,0,-2,-2,0,2,-2,1,-2,-2,0,0,
%U 0,0,0,0,0,-2,1,0,0,0,0,-2,2,0,2,2,0,2,1,0,-2,0,2,0,-2,0,0,4,0,0,0,1,0,0,0,-2,-2,0,0,0,2,-2,0,0,-2
%N Expansion of q^(-1/8) * eta(q^2)^4 / (eta(q) * eta(q^4)) in powers of q.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Number 70 of the 74 eta-quotients listed in Table I of Martin (1996).
%C A030204, A083650 and A138514 are the same except for signs. - _N. J. A. Sloane_, May 07 2010
%H Seiichi Manyama, <a href="/A138514/b138514.txt">Table of n, a(n) for n = 0..10000</a>
%H Y. Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
%H Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</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 Expansion of f(x) * f(-x^2) = psi(-x) * phi(x) = chi(x) * f(-x^2)^2 = psi(x) * phi(-x^2) = f(x)^2 / chi(x) = f(x)^3 / phi(x) = f(-x^2)^3 / psi(-x) = phi(x)^2 / chi(x)^3 = chi(x)^3 * psi(-x)^2 = (f(x)^3 * psi(-x))^(1/2) = (f(-x^2)^3 * phi(x))^(1/2) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
%F Expansion of psi(i * x) * psi(-i * x) in powers x^2 where i^2 = -1 and psi() is a Ramanujan theta function. - _Michael Somos_, Feb 16 2014
%F Euler transform of period 4 sequence [ 1, -3, 1, -2, ...].
%F a(n) = b(8*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = 0 if p == 3, 5, 7 (mod 8) and e odd, b(p^e) = 1 if p == 3 (mod 4) and e even, b(p^e) = (-1)^(e/2) if p == 5 (mod 8) and e even, b(p^e) = e+1 if p == 1 (mod 8) and p = x^2 + 64*y^2, b(p^e) = (-1)^e * (e+1) if p == 1 (mod 8) and p is not of the form x^2 + 64*y^2.
%F a(9*n + 1) = a(n), a(9*n + 4) = a(9*n + 7) = 0. a(n) = (-1)^n * A030204(n) = (-1)^floor((n+1)/2) * A083650(n).
%F G.f.: Product_{k>0} (1 - x^(2*k))^2 * (1 + x^(2*k - 1)).
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (256 t)) = 16 (t/i) f(t) where q = exp(2 Pi i t). - _Michael Somos_, Jun 10 2015
%e G.f. = 1 + x - 2*x^2 - x^3 - 2*x^5 + x^6 + 2*x^9 + x^10 + 2*x^11 - 2*x^12 + ...
%e G.f. = q + q^9 - 2*q^17 - q^25 - 2*q^41 + q^49 + 2*q^73 + q^81 + 2*q^89 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ -x], {x, 0, n}]; (* _Michael Somos_, Jun 10 2015 *)
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^4 / (QPochhammer[ x] QPochhammer[ x^4]), {x, 0, n}]; (* _Michael Somos_, Jun 10 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 / (eta(x + A) * eta(x^4 + A)), n))};
%o (PARI) {a(n) = my(A, p, e); if( n<0, 0, n = 8*n + 1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0, p%8==1, (e+1) * if( qfbclassno( -8 * p) / 4 % 2, (-1)^e, 1), if( e%2==0, (-1)^(e/2 * (p%8==5)))))) };
%o (PARI) {a(n) = if( n<0, 0, n = 8*n + 1; (qfrep([ 1, 0; 0, 64], n) - qfrep([ 4, 2; 2, 17], n))[n])};
%Y Cf. A030204, A083650.
%K sign
%O 0,3
%A _Michael Somos_, Mar 22 2008
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