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Expansion of q^(-5/6) * (eta(q) * eta(q^6)^2 / eta(q^3))^2 in powers of q.
2

%I #10 Aug 02 2018 13:02:10

%S 1,-2,-1,4,-3,0,3,0,1,-2,-2,-4,0,2,3,-4,9,6,-9,0,-6,2,3,4,-7,8,0,-12,

%T -3,-6,6,0,9,0,8,4,2,-6,-5,8,-7,-10,-1,4,5,2,-13,0,9,-8,-2,12,-3,4,0,

%U -4,-16,6,-1,12,10,0,6,0,1,-8,15,-12,0,-6,1,-16,-16

%N Expansion of q^(-5/6) * (eta(q) * eta(q^6)^2 / eta(q^3))^2 in powers of q.

%H G. C. Greubel, <a href="/A258090/b258090.txt">Table of n, a(n) for n = 0..2500</a>

%F Euler transform of period 6 sequence [ -2, -2, 0, -2, -2, -4, ...].

%F G.f.: Product_{k>0} (1 - x^k)^2 * (1 - x^(3*k))^2 * (1 + x^(3*k))^4.

%F -2 * a(n) = A030188(3*n + 2).

%e G.f. = 1 - 2*x - x^2 + 4*x^3 - 3*x^4 + 3*x^6 + x^8 - 2*x^9 - 2*x^10 + ...

%e G.f. = q^5 - 2*q^11 - q^17 + 4*q^23 - 3*q^29 + 3*q^41 + q^53 - 2*q^59 + ...

%t a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^6]^2 / QPochhammer[ x^3])^2, {x, 0, n}];

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^6 + A)^2 / eta(x^3 + A))^2, n))};

%o (PARI) {a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, n = 6*n + 5; A = factor(n); -1/2 * prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0, p==3, (-1)^e, a0 = 1; a1 = y = -sum( x=0, p-1, kronecker( x^3 - x^2 - 4*x + 4, p)); for( i=2, e, x = y*a1 - p*a0; a0 = a1; a1 = x); a1)))};

%o (PARI) q='q+O('q^99); Vec((eta(q)*eta(q^6)^2/eta(q^3))^2) \\ _Altug Alkan_, Aug 02 2018

%Y Cf. A030188.

%K sign

%O 0,2

%A _Michael Somos_, May 19 2015