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A047644 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^19 in powers of x.

%I A047644 #19 Sep 07 2023 02:22:08
%S A047644 1,-19,171,-969,3857,-11286,24206,-34542,14706,83011,-294880,569753,
%T A047644 -680694,220286,1198672,-3502612,5661867,-5571579,791350,9721976,
%U A047644 -23494393,33415357,-29225230,2352751,47086598,-104517176,140834118,-121255530
%N A047644 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^19 in powers of x.
%H A047644 Alois P. Heinz, <a href="/A047644/b047644.txt">Table of n, a(n) for n = 19..10000</a>
%H A047644 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.
%H A047644 H. Gupta, <a href="/A001482/a001482.pdf">On the coefficients of the powers of Dedekind's modular form</a> (annotated and scanned copy)
%F A047644 a(n) = [x^n]( QPochhammer(-x) - 1 )^19. - _G. C. Greubel_, Sep 07 2023
%p A047644 g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*d, d]
%p A047644       [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
%p A047644     end:
%p A047644 b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
%p A047644       (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
%p A047644     end:
%p A047644 a:= n-> b(n, 19):
%p A047644 seq(a(n), n=19..46);  # _Alois P. Heinz_, Feb 07 2021
%t A047644 nmax=46; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] - 1)^19, {x,0,nmax}], x]//Drop[#, 19] & (* _Ilya Gutkovskiy_, Feb 07 2021 *)
%t A047644 With[{k=19}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x,0, 75}], x], k]] (* _G. C. Greubel_, Sep 07 2023 *)
%o A047644 (Magma)
%o A047644 m:=80;
%o A047644 R<x>:=PowerSeriesRing(Integers(), m);
%o A047644 Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(19) )); // _G. C. Greubel_, Sep 07 2023
%o A047644 (SageMath)
%o A047644 from sage.modular.etaproducts import qexp_eta
%o A047644 m=75; k=19;
%o A047644 def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
%o A047644 def A047644_list(prec):
%o A047644     P.<x> = PowerSeriesRing(QQ, prec)
%o A047644     return P( f(k,x) ).list()
%o A047644 a=A047644_list(m); a[k:] # _G. C. Greubel_, Sep 07 2023
%o A047644 (PARI) my(x='x+O('x^35)); Vec((eta(-x)-1)^19) \\ _Joerg Arndt_, Sep 07 2023
%Y A047644 Cf. A001482 - A001488, A001490, A047638 - A047643, A047645 - A047649, A047654, A047655, A341243.
%K A047644 sign
%O A047644 19,2
%A A047644 _N. J. A. Sloane_
%E A047644 Definition and offset edited by _Ilya Gutkovskiy_, Feb 07 2021