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%I #3 Nov 26 2019 15:49:09
%S 1,2,2,3,3,4,4,3,5,3,6,7,4,5,4,8,6,5,7,6,7,8,7,5,8,10,9,4,7,7,9,11,8,
%T 10,5,10,12,7,10,8,10,12,4,10,8,13,15,10,9,5,15,9,12,11,10,12,10,11,
%U 11,12,15,12,6,14,8,11,17,13,12,9,16,17,8,15,10,14
%N Expansion of q^(-13/24) * eta(q^2)^3 * eta(q^3) * eta(q^6) / eta(q)^2 in powers of q.
%F Euler transform of period 6 sequence [2, -1, 1, -1, 2, -3, ...].
%F G.f.: Product_{k>=1} (1 + x^k)^2 * (1 - x^(2*k)) * (1 - x^(3*k)) * (1 - x^(6*k)).
%F Convolution of A033762 and A080995. Convolution of A010054 and A121444.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = (3/2)^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A329955.
%e G.f. = 1 + 2*x + 2*x^2 + 3*x^3 + 3*x^4 + 4*x^5 + 4*x^6 + 3*x^7 + 5*x^8 + ...
%e G.f. = q^13 + 2*q^37 + 2*q^61 + 3*q^85 + 3*q^109 + 4*q^133 + 4*q^157 + ...
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^3 QPochhammer[ x^3] QPochhammer[ x^6] / QPochhammer[ x]^2, {x, 0, n}];
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^6 + A) / eta(x + A)^2, n))};
%Y Cf. A010054, A033762, A080995, A121444, A329955.
%K nonn
%O 0,2
%A _Michael Somos_, Nov 26 2019
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