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%I #25 Sep 07 2023 12:49:58
%S 1,-3,3,-1,-3,6,-6,6,0,-3,6,-9,8,-6,0,0,-6,6,-13,3,-6,3,0,-3,6,-9,6,
%T -3,6,0,6,6,-3,11,0,6,0,9,0,0,0,-3,13,0,0,-6,0,-6,3,-3,-6,0,-15,-6,-3,
%U 0,-6,0,-6,0,-6,-6,0,-11,0,0,-6,0,6,0,6,0,0,0,-3,19,12,-3,0,0,6,6,6,6,0,0,6,0,21,3
%N Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^3 in powers of x.
%H Alois P. Heinz, <a href="/A047655/b047655.txt">Table of n, a(n) for n = 3..10000</a>
%H H. Gupta, <a href="/A001482/a001482.pdf">On the coefficients of the powers of Dedekind's modular form</a> (annotated and scanned copy)
%H H. Gupta, <a href="https://doi.org/10.1112/jlms/s1-39.1.433">On the coefficients of the powers of Dedekind's modular form</a>, J. London Math. Soc., 39 (1964), 433-440.
%F a(n) = [x^n]( QPochhammer(-x) - 1 )^3. - _G. C. Greubel_, Sep 07 2023
%p g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*d, d]
%p [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
%p end:
%p b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
%p (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
%p end:
%p a:= n-> b(n, 3):
%p seq(a(n), n=3..92); # _Alois P. Heinz_, Feb 07 2021
%t nmax=92; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] - 1)^3, {x,0,nmax}], x]//Drop[#, 3] & (* _Ilya Gutkovskiy_, Feb 07 2021 *)
%t With[{k=3}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x,0, 125}], x], k]] (* _G. C. Greubel_, Sep 07 2023 *)
%o (Magma)
%o m:=120;
%o R<x>:=PowerSeriesRing(Integers(), m);
%o Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^3 )); // _G. C. Greubel_, Sep 07 2023
%o (SageMath)
%o from sage.modular.etaproducts import qexp_eta
%o m=125; k=3;
%o def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
%o def A047655_list(prec):
%o P.<x> = PowerSeriesRing(QQ, prec)
%o return P( f(k,x) ).list()
%o a=A047655_list(m); a[k:] # _G. C. Greubel_, Sep 07 2023
%o (PARI) my(x='x+O('x^99)); Vec((eta(-x)-1)^3) \\ _Joerg Arndt_, Sep 07 2023
%Y Cf. A001482 - A001488, A001490, A047265, A047638 - A047649, A047654, A341243.
%K sign
%O 3,2
%A _N. J. A. Sloane_
%E Definition and offset edited by _Ilya Gutkovskiy_, Feb 07 2021
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