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Expansion of (eta(q^4) * eta(q^12) / (eta(q) * eta(q^3)))^2 in powers of q.
4

%I #10 Oct 17 2017 22:18:35

%S 1,2,5,12,22,42,80,136,233,396,636,1020,1622,2496,3822,5808,8642,

%T 12786,18788,27208,39184,56088,79432,111912,156823,217964,301517,

%U 415104,567758,773244,1048616,1414432,1900524,2543940,3389792,4501164,5956430

%N Expansion of (eta(q^4) * eta(q^12) / (eta(q) * eta(q^3)))^2 in powers of q.

%H G. C. Greubel, <a href="/A123647/b123647.txt">Table of n, a(n) for n = 1..1000</a>

%F Euler transform of period 12 sequence [ 2, 2, 4, 0, 2, 4, 2, 0, 4, 2, 2, 0, ...].

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v * (1 + 4*u) * (1 + 4*v).

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (1/16) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A187196. - _Michael Somos_, Sep 02 2015

%F Convolution inverse of A187196. - _Michael Somos_, Sep 02 2015

%F a(n) ~ exp(2*Pi*sqrt(n/3)) / (32 * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Nov 08 2015

%e G.f. = x + 2*x^2 + 5*x^3 + 12*x^4 + 22*x^5 + 42*x^6 + 80*x^7 + 136*x^8 + ...

%t a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^4] QPochhammer[ q^12] / (QPochhammer[ q^] QPochhammer[ q^3]))^2, {q, 0, n}]; (* _Michael Somos_, Sep 02 2015 *)

%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^4 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^3 + A)))^2, n))};

%Y Cf. A187196.

%K nonn

%O 1,2

%A _Michael Somos_, Oct 04 2006