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%I #22 Nov 22 2018 21:50:11
%S 1,-2,-1,4,-4,2,6,-8,1,8,-12,-4,14,-12,4,16,-16,-2,18,-16,-6,24,-24,8,
%T 21,-28,-1,24,-28,-8,30,-32,12,32,-24,4,38,-36,-14,32,-40,12,42,-48,
%U -4,48,-48,-16,43,-42,16,56,-52,2,48,-48,-18,56,-60,16,62,-60,6,64,-56,-24,66,-64,24,48,-72,-8,74,-76,-21,72
%N Expansion of eta(q)^2 * eta(q^4) * eta(q^6)^2 * eta(q^12) / eta(q^3)^2 in powers of q.
%C Number 26 of the 74 eta-quotients listed in Table I of Martin (1996).
%H G. C. Greubel, <a href="/A113421/b113421.txt">Table of n, a(n) for n = 1..1000</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 Euler transform of period 12 sequence [-2, -2, 0, -3, -2, -2, -2, -3, 0, -2, -2, -4, ...].
%F G.f.: Sum_{k>0} (3*k - 2) * x^(3*k - 2) / (1 + x^(6*k - 4)) - (3*k - 1) * x^(3*k - 1) / (1 + x^(6*k - 2)).
%F G.f.: Sum_{k>0} -(-1)^k * x^(2*k - 1) * (1 - x^(2*k - 1))^2 * (1 - x^(4*k - 2)) / (1 - x^(6*k - 3))^2.
%F a(n) is multiplicative with a(2^e) = (-2)^e, a(3^e) = (-1)^e, a(p^e) = (x^(e+1) - y^(e+1)) / (x - y) where x = p * Kronecker( -3, p) and y = (-1)^[p/2].
%e G.f. = q - 2*q^2 - q^3 + 4*q^4 - 4*q^5 + 2*q^6 + 6*q^7 - 8*q^8 + q^9 + 8*q^10 + ...
%t a[ n_] := SeriesCoefficient[ q QPochhammer[ q]^2 QPochhammer[ q^4] QPochhammer[ q^6]^2 QPochhammer[ q^12] / QPochhammer[ q^3]^2, {q, 0, n}]; (* _Michael Somos_, Jul 09 2015 *)
%o (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * d * kronecker( -3, d) * (-1)^(n / d \ 2)))};
%o (PARI) {a(n) = my(A, p, e, t); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, (-2)^e, p==3, (-1)^e, t = (-1)^(p\2); p *= kronecker( -3, p); (p^(e+1) - t^(e+1)) / (p - t))))};
%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A)^2 * eta(x^12 + A) / eta(x^3 + A)^2, n))};
%K sign,mult
%O 1,2
%A _Michael Somos_, Oct 29 2005
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