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%I #11 Sep 08 2022 08:46:05
%S 1,64,1236,4096,-57450,79104,64232,262144,-66627,-3676800,2464572,
%T 5062656,8032766,4110848,-71008200,16777216,71112402,-4264128,
%U 136337060,-235315200,79390752,157732608,-1186563144,324009984,2079799375,514097024,-2052934200,263094272
%N Expansion of (2 * eta(q^2)^24 - eta(q)^16 * eta(q^4)^8)^3 / (eta(q)^4 * eta(q^2) * eta(q^4)^6)^4 in powers of q.
%H G. C. Greubel, <a href="/A226086/b226086.txt">Table of n, a(n) for n = 1..2500</a>
%F a(n) is multiplicative with a(2^n) = 64^n, a(p^e) = a(p) * a(p^(e-1)) - p^13 * a(p^(e-2)) if p>2.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (2 t)) = 128 (t/i)^14 f(t) where q = exp(2 Pi i t).
%F a(2*n) = 64 * a(n).
%e G.f. = q + 64*q^2 + 1236*q^3 + 4096*q^4 - 57450*q^5 + 79104*q^6 + 64232*q^7 + ...
%t eta[q_] := q^(1/24)*QPochhammer[q]; Drop[CoefficientList[Series[(2* eta[q^2]^24 - eta[q]^16*eta[q^4]^8)^3/(eta[q]^4*eta[q^2]*eta[q^4]^6)^4, {q, 0, 50}], q], 1] (* _G. C. Greubel_, Aug 09 2018 *)
%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (2 * eta(x^2 + A)^24 - eta(x + A)^16 * eta(x^4 + A)^8)^3 / (eta(x + A)^4 * eta(x^2 + A) * eta(x^4 + A)^6)^4, n))};
%o (Sage) A = CuspForms( Gamma1(2), 14, prec=29) . basis(); A[0] + 64*A[1];
%o (Magma) A := Basis( CuspForms( Gamma1(2), 14), 29); A[1] + 64*A[2];
%K sign,mult
%O 1,2
%A _Michael Somos_, May 25 2013
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