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A001486 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^8 in powers of x.

%I A001486 M4502 N1906 #25 Sep 05 2023 01:21:22
%S A001486 1,-8,28,-56,62,0,-148,328,-419,280,140,-728,1232,-1336,848,224,-1582,
%T A001486 2688,-3072,2408,-742,-1568,3836,-5264,5306,-3744,924,2576,-5686,7792,
%U A001486 -8092,6272,-2751,-1848,6008,-9296,10556,-9800,6692,-2240,-3206,8168,-11524
%N A001486 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^8 in powers of x.
%D A001486 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001486 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001486 Alois P. Heinz, <a href="/A001486/b001486.txt">Table of n, a(n) for n = 8..10000</a>
%H A001486 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 A001486 H. Gupta, <a href="/A001482/a001482.pdf">On the coefficients of the powers of Dedekind's modular form</a> (annotated and scanned copy)
%F A001486 a(n) = [x^n]( QPochhammer(-x) - 1 )^8. - _G. C. Greubel_, Sep 04 2023
%p A001486 g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*d, d]
%p A001486       [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
%p A001486     end:
%p A001486 b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
%p A001486       (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
%p A001486     end:
%p A001486 a:= n-> b(n, 8):
%p A001486 seq(a(n), n=8..50);  # _Alois P. Heinz_, Feb 07 2021
%t A001486 nmax=50; CoefficientList[Series[(Product[(1 -(-x)^j), {j,nmax}] -1)^8, {x,0,nmax}], x]//Drop[#,8] & (* _Ilya Gutkovskiy_, Feb 07 2021 *)
%t A001486 Drop[CoefficientList[Series[(QPochhammer[-x] -1)^8, {x,0,102}], x], 8] (* _G. C. Greubel_, Sep 04 2023 *)
%o A001486 (Magma)
%o A001486 m:=102;
%o A001486 R<x>:=PowerSeriesRing(Integers(), m);
%o A001486 Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^8 )); // _G. C. Greubel_, Sep 04 2023
%o A001486 (SageMath)
%o A001486 from sage.modular.etaproducts import qexp_eta
%o A001486 m=100; k=8;
%o A001486 def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
%o A001486 def A001486_list(prec):
%o A001486     P.<x> = PowerSeriesRing(QQ, prec)
%o A001486     return P( f(k,x) ).list()
%o A001486 a=A001486_list(m); a[k:] # _G. C. Greubel_, Sep 04 2023
%o A001486 (PARI) my(N=55,x='x+O('x^N)); Vec((eta(-x)-1)^8) \\ _Joerg Arndt_, Sep 05 2023
%Y A001486 Cf. A001482 - A001485, A001487, A001488, A047638 - A047649, A047654, A047655, A341243.
%K A001486 sign
%O A001486 8,2
%A A001486 _N. J. A. Sloane_
%E A001486 Definition and offset edited by _Ilya Gutkovskiy_, Feb 07 2021