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%I #6 Apr 26 2015 11:40:40
%S 1,0,-2,0,0,2,0,0,0,-3,0,0,0,0,4,0,0,0,0,0,-4,0,0,0,0,0,0,5,0,0,0,0,0,
%T 0,0,-6,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,-7,0,0,0,0,0,0,0,0,0,0,8,
%U 0,0,0,0,0,0,0,0,0,0,0,-8,0,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,0,-10
%N Expansion of q^(-1/8) * (eta(q^3) - eta(q)^3) / 3 in powers of q.
%F G.f.: Sum_{k>1} (-1)^k * floor((2*k)/3) * x^((k^2 - k)/2).
%e G.f. = x - 2*x^3 + 2*x^6 - 3*x^10 + 4*x^15 - 4*x^21 + 5*x^28 - 6*x^36 + 6*x^45 - ...
%e G.f. = q^9 - 2*q^25 + 2*q^49 - 3*q^81 + 4*q^121 - 4*q^169 + 5*q^225 - 6*q^289 + ...
%t a[ n_] := SeriesCoefficient[ (QPochhammer[ x^3] - QPochhammer[ x]^3) / 3, {x, 0, n}]; (* _Michael Somos_, Apr 26 2015 *)
%t a[ n_] := Which[ n < 0, 0, EvenQ[ Length @ Divisors [8 n + 1]], 0, True, With[ {k = Sqrt[8 n + 1] + 1}, Quotient[k, 3] (-1)^Quotient[k, 2]]]; (* _Michael Somos_, Apr 26 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); 1/3 * polcoeff( eta(x^3 + A) - eta(x + A)^3, n))};
%o (PARI) {a(n) = if( n<0, 0, if( issquare(8*n + 1, &n), ((n+1) \ 3) * (-1)^((n+1) / 2)))};
%K sign
%O 1,3
%A _Michael Somos_, Jun 19 2005
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