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A047654 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^2 in powers of x.

%I A047654 #23 Sep 07 2023 11:09:29
%S A047654 1,-2,1,0,-2,2,-2,2,1,0,2,-2,3,0,2,0,0,2,-2,0,-2,2,-1,0,0,-2,-2,-2,1,
%T A047654 -2,0,-2,-2,0,2,0,-2,0,-2,0,0,0,1,2,0,0,2,0,2,0,1,2,0,-2,2,2,0,2,0,2,
%U A047654 0,2,2,0,-4,0,0,2,1,-2,0,-2,0,0,0,0,2,-4,1,0,0,-2,-2,-2,-2,0,0,-2,0,2,-2,2,-2
%N A047654 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^2 in powers of x.
%H A047654 Alois P. Heinz, <a href="/A047654/b047654.txt">Table of n, a(n) for n = 2..10000</a>
%H A047654 H. Gupta, <a href="/A001482/a001482.pdf">On the coefficients of the powers of Dedekind's modular form</a> (annotated and scanned copy)
%H A047654 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 A047654 a(n) = [x^n]( QPochhammer(-x) - 1 )^2. - _G. C. Greubel_, Sep 07 2023
%p A047654 g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*d, d]
%p A047654       [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
%p A047654     end:
%p A047654 b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
%p A047654       (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
%p A047654     end:
%p A047654 a:= n-> b(n, 2):
%p A047654 seq(a(n), n=2..94);  # _Alois P. Heinz_, Feb 07 2021
%t A047654 nmax=94; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] - 1)^2, {x, 0, nmax}], x]//Drop[#, 2] & (* _Ilya Gutkovskiy_, Feb 07 2021 *)
%t A047654 With[{k=2}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x,0, 125}], x], k]] (* _G. C. Greubel_, Sep 07 2023 *)
%o A047654 (PARI) seq(n)={Vec((prod(j=1, n, 1-(-x)^j + O(x^n)) - 1)^2)} \\ _Andrew Howroyd_, Feb 07 2021
%o A047654 (Magma)
%o A047654 m:=120;
%o A047654 R<x>:=PowerSeriesRing(Integers(), m);
%o A047654 Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^2 )); // _G. C. Greubel_, Sep 07 2023
%o A047654 (SageMath)
%o A047654 from sage.modular.etaproducts import qexp_eta
%o A047654 m=125; k=2;
%o A047654 def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
%o A047654 def A047654_list(prec):
%o A047654     P.<x> = PowerSeriesRing(QQ, prec)
%o A047654     return P( f(k,x) ).list()
%o A047654 a=A047654_list(m); a[k:] # _G. C. Greubel_, Sep 07 2023
%Y A047654 Cf. A001482 - A001488, A001490, A047265, A047638 - A047649, A047655, A341243.
%K A047654 sign
%O A047654 2,2
%A A047654 _N. J. A. Sloane_
%E A047654 Definition and offset edited by _Ilya Gutkovskiy_, Feb 07 2021