%I M3263 N1317 #30 Sep 04 2023 06:07:55

%S 1,-4,6,-4,-3,12,-16,16,-6,-8,18,-28,26,-20,2,12,-23,32,-36,28,-6,4,

%T 22,-20,39,-32,32,-12,2,16,-12,24,-40,28,-34,0,-6,-16,0,-40,6,-36,26,

%U -32,-5,0,-20,8,-16,12,-10,40,-22,12,14,12,45,16,38,4,12,0,34,8,38,12,-24,44,2,16

%N Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^4 in powers of x.

%D H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.

%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="/A001482/b001482.txt">Table of n, a(n) for n = 4..10000</a>

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

%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, 4):

%p seq(a(n), n=4..73); # _Alois P. Heinz_, Feb 07 2021

%t nmax = 73; CoefficientList[Series[(Product[(1 - (-x)^j), {j, 1, nmax}] - 1)^4, {x, 0, nmax}], x] // Drop[#, 4] & (* _Ilya Gutkovskiy_, Feb 07 2021 *)

%t Drop[CoefficientList[Series[(1 -QPochhammer[-x])^4, {x,0,100}], x], 4] (* _G. C. Greubel_, Sep 04 2023 *)

%o (Magma)

%o m:=102;

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

%o Coefficients(R!( (1 - (&*[1-(-x)^j: j in [1..m+2]]))^4 )); // _G. C. Greubel_, Sep 04 2023

%o (SageMath)

%o m=100

%o def f4(x): return (1 - product( (1+x^j)*(1-x^(2*j))/(1+x^(2*j)) for j in range(1,m+2) ) )^4

%o def A001482_list(prec):

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

%o return P( f4(x) ).list()

%o a=A001482_list(m); a[4:] # _G. C. Greubel_, Sep 04 2023

%o (PARI) my(N=70,x='x+O('x^N)); Vec((eta(-x)-1)^4) \\ _Joerg Arndt_, Sep 04 2023

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

%K sign

%O 4,2

%A _N. J. A. Sloane_

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