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%I #30 Oct 18 2023 04:36:54
%S 1,-1,-1,0,0,1,0,0,1,1,0,0,-2,0,-2,0,1,0,0,0,0,-1,2,0,0,0,2,0,0,0,0,0,
%T 0,-1,0,0,-2,-1,0,0,-1,0,2,0,0,0,0,0,0,2,-2,0,0,0,2,0,1,2,-1,0,0,0,0,
%U 0,-2,-1,0,0,0,0,-2,0,0,0,0,0,0,0,0,0,0,0
%N Expansion of Product_{i >= 1} (1 - q^i)*(1 - q^(7*i)).
%C Number 56 of the 74 eta-quotients listed in Table I of Martin (1996).
%H Seiichi Manyama, <a href="/A002655/b002655.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>
%F Expansion of q^(-1/3) * eta(q) * eta(q^7) in powers of q.
%F Euler transform of period 7 sequence [ -1, -1, -1, -1, -1, -1, -2, ...]. - _Michael Somos_, Dec 06 2004
%F Given g.f. A(x), B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^2*w + 2 * u*w^2 - v^3 - _Michael Somos_, Dec 06 2004
%F G.f.: Product_{k>0} (1 - x^k) * (1 - x^(7*k)).
%F a(n) = b(3*n + 1) where b(n) is multiplicative and b(p^(2*e)) = (-1)^e if p=2, b(p^e) = 0^e if p = 3, b(p^e) = (-1)^e if p = 7, b(p^e) = (1 + (-1)^e) / 2 if p == 3, 5, 6 (mod 7), else p == 1, 2, 4 (mod 7) and p = y^2 + 7*x^2 when b(p^(2*e)) = (-1)^e if x*y not divisible by 3, b(p^e) = e+1 if x divisible by 3 or (e+1) * (-1)^e if y divisible by 3 . - _Michael Somos_, May 28 2005
%F a(2*n) = A160806(n). a(4*n + 3) = 0. a(4*n + 1) = -a(n). a(7*n + 3) = a(7*n + 4) = a(7*n + 6) = 0.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (63 t)) = 63^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - _Michael Somos_, May 17 2015
%e G.f. = 1 - x - x^2 + x^5 + x^8 + x^9 - 2*x^12 - 2*x^14 + x^16 - x^21 + 2*x^22 + ...
%e G.f. = q - q^4 - q^7 + q^16 + q^25 + q^28 - 2*q^37 - 2*q^43 + q^49 - q^64 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^7], {x, 0, n}]; (* _Michael Somos_, Feb 22 2015 *)
%o (PARI) {a(n) = if( n<0, 0, n = 3*n + 1; (qfrep( [2, 1; 1, 32], n, 1) - qfrep( [8, 1; 1, 8], n, 1))[n])}; /* _Michael Somos_, May 28 2005 */
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^7 + A), n))}; /* _Michael Somos_, May 28 2005 */
%o (PARI) {a(n) = my(A, p, e, x, y); if( n<0, 0, n = 3*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, real(I^e), p==3, 0^e, p==7, (-1)^e, kronecker( p, 7)==-1, !(e%2), for( x=0, sqrtint(p\7), if( issquare(p - 7*x^2, &y), y = if( x%3 && y%3, real(I^e), (e+1) * if( x%3, (-1)^e,1)); break)); y)))}; /* _Michael Somos_, May 28 2005 */
%o (Magma) Basis( CuspForms( Gamma1(63), 1), 242) [1]; /* _Michael Somos_, May 17 2015 */
%Y Cf. A160806.
%K sign
%O 0,13
%A _N. J. A. Sloane_
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