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

%I #4 Nov 26 2019 15:14:02

%S 1,-1,-2,-2,3,8,0,-2,-10,-4,2,4,10,-8,-4,0,7,12,4,-2,-16,-16,4,8,0,-7,

%T -4,-2,10,24,8,-2,-26,0,2,8,12,-16,-8,-8,10,12,0,-6,-20,-16,4,8,26,-7,

%U -10,0,16,40,0,-4,-20,-24,6,4,0,-16,-12,-8,15,24,8,-6

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

%F Euler transform of period 6 sequence [-1, -2, -4, -2, -1, -3, ...].

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

%F Convolution of A030206 and A195848.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 1990656^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A329958.

%F a(3*n) = A224822(n). a(3*n + 1) = -A329956(n). a(3*n + 2) = -2*A329957(n). a(6*n) = A028967(n).

%e G.f. = 1 - x - 2*x^2 - 2*x^3 + 3*x^4 + 8*x^5 - 2*x^7 - 10*x^8 - 4*x^9 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^2] QPochhammer[ x^3]^3 / QPochhammer[ x^6]^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^2 + A) * eta(x^3 + A)^3 / eta(x^6 + A)^2, n))};

%Y Cf. A028967, A030206, A195848, A224822, A329956, A329957.

%K sign

%O 0,3

%A _Michael Somos_, Nov 26 2019