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A001484 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^6 in powers of x.

%I A001484 M4107 N1704 #28 Sep 04 2023 06:05:02
%S A001484 1,-6,15,-20,9,24,-65,90,-75,6,90,-180,220,-180,66,110,-264,360,-365,
%T A001484 264,-66,-178,375,-510,496,-414,180,60,-330,570,-622,582,-390,220,96,
%U A001484 -300,621,-630,705,-492,300,0,-235,420,-570,594,-735,420,-420,-120,219,-586,360
%N A001484 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^6 in powers of x.
%D A001484 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001484 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001484 Robert Israel, <a href="/A001484/b001484.txt">Table of n, a(n) for n = 6..10000</a>
%H A001484 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 A001484 H. Gupta, <a href="/A001482/a001482.pdf">On the coefficients of the powers of Dedekind's modular form</a> (annotated and scanned copy)
%F A001484 a(n) = [x^n] ( QPochhammer(-x) - 1 )^6. - _G. C. Greubel_, Sep 04 2023
%p A001484 N:= 100:
%p A001484 S:= series((mul(1-(-x)^j,j=1..N)-1)^6,x,N+1):
%p A001484 seq(coeff(S,x,j),j=6..N); # _Robert Israel_, Feb 05 2019
%t A001484 Drop[CoefficientList[Series[(QPochhammer[-x] -1)^6, {x,0,102}], x], 6] (* _G. C. Greubel_, Sep 04 2023 *)
%o A001484 (Magma)
%o A001484 m:=102;
%o A001484 R<x>:=PowerSeriesRing(Integers(), m);
%o A001484 Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^6 )); // _G. C. Greubel_, Sep 04 2023
%o A001484 (SageMath)
%o A001484 m=100; k=6;
%o A001484 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 A001484 def A001484_list(prec):
%o A001484     P.<x> = PowerSeriesRing(QQ, prec)
%o A001484     return P( f(k,x) ).list()
%o A001484 a=A001484_list(m); a[k:] # _G. C. Greubel_, Sep 04 2023
%o A001484 (PARI) my(N=70,x='x+O('x^N)); Vec((eta(-x)-1)^6) \\ _Joerg Arndt_, Sep 04 2023
%Y A001484 Cf. A001482, A001483, A001485 - A001488, A047638 - A047649, A047654, A047655, A341243.
%K A001484 sign
%O A001484 6,2
%A A001484 _N. J. A. Sloane_
%E A001484 Edited by _Robert Israel_, Feb 05 2019