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A047647 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^22 in powers of x.

%I A047647 #21 Sep 05 2023 08:15:56
%S A047647 1,-22,231,-1540,7293,-25872,69971,-140822,183711,-25102,-634480,
%T A047647 2027804,-3817814,4439116,-919600,-9829270,27660479,-44779042,
%U A047647 43632974,-1898820,-92518261,219961214,-313463842,267448104,15757973,-547042056,1173033400
%N A047647 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^22 in powers of x.
%H A047647 Alois P. Heinz, <a href="/A047647/b047647.txt">Table of n, a(n) for n = 22..10000</a>
%H A047647 H. Gupta, <a href="/A001482/a001482.pdf">On the coefficients of the powers of Dedekind's modular form</a> (annotated and scanned copy)
%H A047647 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 A047647 a(n) = [x^n]( QPochhammer(-x) - 1 )^22. - _G. C. Greubel_, Sep 05 2023
%p A047647 g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*d, d]
%p A047647       [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
%p A047647     end:
%p A047647 b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
%p A047647       (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
%p A047647     end:
%p A047647 a:= n-> b(n, 22):
%p A047647 seq(a(n), n=22..48);  # _Alois P. Heinz_, Feb 07 2021
%t A047647 nmax=48; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] - 1)^22, {x, 0, nmax}], x]//Drop[#, 22] & (* _Ilya Gutkovskiy_, Feb 07 2021 *)
%t A047647 With[{k=22}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x,0, 75}], x], k]] (* _G. C. Greubel_, Sep 05 2023 *)
%o A047647 (Magma)
%o A047647 m:=75;
%o A047647 R<x>:=PowerSeriesRing(Integers(), m);
%o A047647 Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(22) )); // _G. C. Greubel_, Sep 05 2023
%o A047647 (SageMath)
%o A047647 from sage.modular.etaproducts import qexp_eta
%o A047647 m=75; k=22;
%o A047647 def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
%o A047647 def A047647_list(prec):
%o A047647     P.<x> = PowerSeriesRing(QQ, prec)
%o A047647     return P( f(k,x) ).list()
%o A047647 a=A047647_list(m); a[k:] # _G. C. Greubel_, Sep 05 2023
%o A047647 (PARI) my(N=55, x='x+O('x^N)); Vec((eta(-x)-1)^22) \\ _Michel Marcus_, Sep 05 2023
%Y A047647 Cf. A001482 - A001488, A001490, A047638 - A047646, A047648, A047649, A047654, A047655, A341243.
%K A047647 sign
%O A047647 22,2
%A A047647 _N. J. A. Sloane_
%E A047647 Definition and offset edited by _Ilya Gutkovskiy_, Feb 07 2021