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Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^8 in powers of x.
(Formerly M4502 N1906)
5

%I M4502 N1906 #25 Sep 05 2023 01:21:22

%S 1,-8,28,-56,62,0,-148,328,-419,280,140,-728,1232,-1336,848,224,-1582,

%T 2688,-3072,2408,-742,-1568,3836,-5264,5306,-3744,924,2576,-5686,7792,

%U -8092,6272,-2751,-1848,6008,-9296,10556,-9800,6692,-2240,-3206,8168,-11524

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

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Alois P. Heinz, <a href="/A001486/b001486.txt">Table of n, a(n) for n = 8..10000</a>

%H 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 H. Gupta, <a href="/A001482/a001482.pdf">On the coefficients of the powers of Dedekind's modular form</a> (annotated and scanned copy)

%F a(n) = [x^n]( QPochhammer(-x) - 1 )^8. - _G. C. Greubel_, Sep 04 2023

%p g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*d, d]

%p [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)

%p end:

%p b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),

%p (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))

%p end:

%p a:= n-> b(n, 8):

%p seq(a(n), n=8..50); # _Alois P. Heinz_, Feb 07 2021

%t nmax=50; CoefficientList[Series[(Product[(1 -(-x)^j), {j,nmax}] -1)^8, {x,0,nmax}], x]//Drop[#,8] & (* _Ilya Gutkovskiy_, Feb 07 2021 *)

%t Drop[CoefficientList[Series[(QPochhammer[-x] -1)^8, {x,0,102}], x], 8] (* _G. C. Greubel_, Sep 04 2023 *)

%o (Magma)

%o m:=102;

%o R<x>:=PowerSeriesRing(Integers(), m);

%o Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^8 )); // _G. C. Greubel_, Sep 04 2023

%o (SageMath)

%o from sage.modular.etaproducts import qexp_eta

%o m=100; k=8;

%o def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k

%o def A001486_list(prec):

%o P.<x> = PowerSeriesRing(QQ, prec)

%o return P( f(k,x) ).list()

%o a=A001486_list(m); a[k:] # _G. C. Greubel_, Sep 04 2023

%o (PARI) my(N=55,x='x+O('x^N)); Vec((eta(-x)-1)^8) \\ _Joerg Arndt_, Sep 05 2023

%Y Cf. A001482 - A001485, A001487, A001488, A047638 - A047649, A047654, A047655, A341243.

%K sign

%O 8,2

%A _N. J. A. Sloane_

%E Definition and offset edited by _Ilya Gutkovskiy_, Feb 07 2021