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A001483 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^5 in powers of x.

%I A001483 M3791 N1546 #29 Sep 04 2023 06:09:00
%S A001483 1,-5,10,-10,0,19,-35,40,-25,-10,45,-75,80,-60,15,45,-85,115,-115,90,
%T A001483 -21,-35,95,-130,135,-135,70,-35,-65,105,-146,120,-150,90,-65,-25,90,
%U A001483 -115,150,-125,130,-45,80,35,-5,160,-110,170,-85,95,25,50,0,-60,95,-116,120,-135
%N A001483 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^5 in powers of x.
%D A001483 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001483 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001483 Alois P. Heinz, <a href="/A001483/b001483.txt">Table of n, a(n) for n = 5..10000</a>
%H A001483 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 A001483 H. Gupta, <a href="/A001482/a001482.pdf">On the coefficients of the powers of Dedekind's modular form</a> (annotated and scanned copy)
%F A001483 a(n) = [x^n]( QPochhammer(-x) - 1 )^5. - _G. C. Greubel_, Sep 04 2023
%p A001483 g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*d, d]
%p A001483       [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
%p A001483     end:
%p A001483 b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
%p A001483       (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
%p A001483     end:
%p A001483 a:= n-> b(n, 5):
%p A001483 seq(a(n), n=5..62);  # _Alois P. Heinz_, Feb 07 2021
%t A001483 nmax = 62; CoefficientList[Series[(Product[(1 - (-x)^j), {j, 1, nmax}] - 1)^5, {x, 0, nmax}], x] // Drop[#, 5] & (* _Ilya Gutkovskiy_, Feb 07 2021 *)
%t A001483 Drop[CoefficientList[Series[(QPochhammer[-x] -1)^5, {x,0,102}], x], 5] (* _G. C. Greubel_, Sep 04 2023 *)
%o A001483 (Magma)
%o A001483 m:=102;
%o A001483 R<x>:=PowerSeriesRing(Integers(), m);
%o A001483 Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^5 )); // _G. C. Greubel_, Sep 04 2023
%o A001483 (SageMath)
%o A001483 m=100; k=5;
%o A001483 def f(k,x): return (-1 + product( (1+x^j)*(1-x^(2*j))/(1+x^(2*j)) for j in range(1,m+2) ) )^k
%o A001483 def A001483_list(prec):
%o A001483     P.<x> = PowerSeriesRing(QQ, prec)
%o A001483     return P( f(k,x) ).list()
%o A001483 a=A001483_list(m); a[k:] # _G. C. Greubel_, Sep 04 2023
%o A001483 (PARI) my(N=70,x='x+O('x^N)); Vec((eta(-x)-1)^5) \\ _Joerg Arndt_, Sep 04 2023
%Y A001483 Cf. A001482, A001484 - A001488, A047638 - A047649, A047654, A047655, A341243.
%K A001483 sign
%O A001483 5,2
%A A001483 _N. J. A. Sloane_
%E A001483 Definition and offset edited by _Ilya Gutkovskiy_, Feb 07 2021