A001484 - OEIS

%I M4107 N1704 #28 Sep 04 2023 06:05:02

%S 1,-6,15,-20,9,24,-65,90,-75,6,90,-180,220,-180,66,110,-264,360,-365,

%T 264,-66,-178,375,-510,496,-414,180,60,-330,570,-622,582,-390,220,96,

%U -300,621,-630,705,-492,300,0,-235,420,-570,594,-735,420,-420,-120,219,-586,360

%N Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^6 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 Robert Israel, <a href="/A001484/b001484.txt">Table of n, a(n) for n = 6..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 )^6. - _G. C. Greubel_, Sep 04 2023

%p N:= 100:

%p S:= series((mul(1-(-x)^j,j=1..N)-1)^6,x,N+1):

%p seq(coeff(S,x,j),j=6..N); # _Robert Israel_, Feb 05 2019

%t Drop[CoefficientList[Series[(QPochhammer[-x] -1)^6, {x,0,102}], x], 6] (* _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)^6 )); // _G. C. Greubel_, Sep 04 2023

%o (SageMath)

%o m=100; k=6;

%o 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 def A001484_list(prec):

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

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

%o a=A001484_list(m); a[k:] # _G. C. Greubel_, Sep 04 2023

%o (PARI) my(N=70,x='x+O('x^N)); Vec((eta(-x)-1)^6) \\ _Joerg Arndt_, Sep 04 2023

%Y Cf. A001482, A001483, A001485 - A001488, A047638 - A047649, A047654, A047655, A341243.

%K sign

%O 6,2

%A _N. J. A. Sloane_

%E Edited by _Robert Israel_, Feb 05 2019