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A047643 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^18 in powers of x.

%I A047643 #21 Sep 07 2023 02:22:11
%S A047643 1,-18,153,-816,3042,-8262,16098,-19278,-1377,72556,-203184,339030,
%T A047643 -326961,-53244,940050,-2147916,2975391,-2293488,-911369,6616332,
%U A047643 -12906162,15883884,-10936899,-4660974,28758849,-52660134,62518248,-44501988,-7465464,84565242
%N A047643 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^18 in powers of x.
%H A047643 Alois P. Heinz, <a href="/A047643/b047643.txt">Table of n, a(n) for n = 18..10000</a>
%H A047643 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 A047643 H. Gupta, <a href="/A001482/a001482.pdf">On the coefficients of the powers of Dedekind's modular form</a> (annotated and scanned copy)
%F A047643 a(n) = [x^n]( QPochhammer(-x) - 1 )^18. - _G. C. Greubel_, Sep 07 2023
%p A047643 g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*d, d]
%p A047643       [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
%p A047643     end:
%p A047643 b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
%p A047643       (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
%p A047643     end:
%p A047643 a:= n-> b(n, 18):
%p A047643 seq(a(n), n=18..47);  # _Alois P. Heinz_, Feb 07 2021
%t A047643 nmax=47; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] -1)^18, {x,0,nmax}], x]//Drop[#, 18] & (* _Ilya Gutkovskiy_, Feb 07 2021 *)
%t A047643 With[{k=18}, Drop[CoefficientList[Series[(QPochhammer[-x]-1)^k, {x,0, 75}], x], k]] (* _G. C. Greubel_, Sep 07 2023 *)
%o A047643 (Magma)
%o A047643 m:=80;
%o A047643 R<x>:=PowerSeriesRing(Integers(), m);
%o A047643 Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(18) )); // _G. C. Greubel_, Sep 07 2023
%o A047643 (SageMath)
%o A047643 from sage.modular.etaproducts import qexp_eta
%o A047643 m=75; k=18;
%o A047643 def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
%o A047643 def A047643_list(prec):
%o A047643     P.<x> = PowerSeriesRing(QQ, prec)
%o A047643     return P( f(k,x) ).list()
%o A047643 a=A047643_list(m); a[k:] # _G. C. Greubel_, Sep 07 2023
%o A047643 (PARI) my(x='x+O('x^35)); Vec((eta(-x)-1)^18) \\ _Joerg Arndt_, Sep 07 2023
%Y A047643 Cf. A001482 - A001488, A001490, A047638 - A047642, A047644 - A047649, A047654, A047655, A341243.
%K A047643 sign
%O A047643 18,2
%A A047643 _N. J. A. Sloane_
%E A047643 Definition and offset edited by _Ilya Gutkovskiy_, Feb 07 2021