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A047641 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^16 in powers of x.

%I A047641 #20 Sep 07 2023 02:22:18
%S A047641 1,-16,120,-560,1804,-4128,6312,-3920,-10530,42208,-82752,99584,
%T A047641 -39460,-141200,422568,-673936,660941,-144720,-938840,2301568,
%U A047641 -3257188,2916592,-628040,-3492160,8217536,-11341568,10408280,-3885040,-7668720,21033408
%N A047641 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^16 in powers of x.
%H A047641 Alois P. Heinz, <a href="/A047641/b047641.txt">Table of n, a(n) for n = 16..10000</a>
%H A047641 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 A047641 H. Gupta, <a href="/A001482/a001482.pdf">On the coefficients of the powers of Dedekind's modular form</a> (annotated and scanned copy)
%F A047641 a(n) = [x^n]( QPochhammer(-x) - 1 )^16. - _G. C. Greubel_, Sep 07 2023
%p A047641 g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*d, d]
%p A047641       [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
%p A047641     end:
%p A047641 b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
%p A047641       (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
%p A047641     end:
%p A047641 a:= n-> b(n, 16):
%p A047641 seq(a(n), n=16..45);  # _Alois P. Heinz_, Feb 07 2021
%t A047641 nmax=45; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] -1)^16, {x,0,nmax}], x]//Drop[#, 16] & (* _Ilya Gutkovskiy_, Feb 07 2021 *)
%t A047641 With[{k=16}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x,0, 75}], x], k]] (* _G. C. Greubel_, Sep 07 2023 *)
%o A047641 (Magma)
%o A047641 m:=80;
%o A047641 R<x>:=PowerSeriesRing(Integers(), m);
%o A047641 Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(16) )); // _G. C. Greubel_, Sep 07 2023
%o A047641 (SageMath)
%o A047641 from sage.modular.etaproducts import qexp_eta
%o A047641 m=75; k=16;
%o A047641 def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
%o A047641 def A047641_list(prec):
%o A047641     P.<x> = PowerSeriesRing(QQ, prec)
%o A047641     return P( f(k,x) ).list()
%o A047641 a=A047641_list(m); a[k:] # _G. C. Greubel_, Sep 07 2023
%o A047641 (PARI) my(x='x+O('x^40)); Vec((eta(-x)-1)^16) \\ _Joerg Arndt_, Sep 07 2023
%Y A047641 Cf. A001482 - A001488, A001490, A047638 - A047640, A047642 - A047649, A047654, A047655, A341243.
%K A047641 sign
%O A047641 16,2
%A A047641 _N. J. A. Sloane_
%E A047641 Definition and offset edited by _Ilya Gutkovskiy_, Feb 07 2021