login
Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^2 in powers of x.
25

%I #23 Sep 07 2023 11:09:29

%S 1,-2,1,0,-2,2,-2,2,1,0,2,-2,3,0,2,0,0,2,-2,0,-2,2,-1,0,0,-2,-2,-2,1,

%T -2,0,-2,-2,0,2,0,-2,0,-2,0,0,0,1,2,0,0,2,0,2,0,1,2,0,-2,2,2,0,2,0,2,

%U 0,2,2,0,-4,0,0,2,1,-2,0,-2,0,0,0,0,2,-4,1,0,0,-2,-2,-2,-2,0,0,-2,0,2,-2,2,-2

%N Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^2 in powers of x.

%H Alois P. Heinz, <a href="/A047654/b047654.txt">Table of n, a(n) for n = 2..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)

%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.

%F a(n) = [x^n]( QPochhammer(-x) - 1 )^2. - _G. C. Greubel_, Sep 07 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, 2):

%p seq(a(n), n=2..94); # _Alois P. Heinz_, Feb 07 2021

%t nmax=94; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] - 1)^2, {x, 0, nmax}], x]//Drop[#, 2] & (* _Ilya Gutkovskiy_, Feb 07 2021 *)

%t With[{k=2}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x,0, 125}], x], k]] (* _G. C. Greubel_, Sep 07 2023 *)

%o (PARI) seq(n)={Vec((prod(j=1, n, 1-(-x)^j + O(x^n)) - 1)^2)} \\ _Andrew Howroyd_, Feb 07 2021

%o (Magma)

%o m:=120;

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

%o Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^2 )); // _G. C. Greubel_, Sep 07 2023

%o (SageMath)

%o from sage.modular.etaproducts import qexp_eta

%o m=125; k=2;

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

%o def A047654_list(prec):

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

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

%o a=A047654_list(m); a[k:] # _G. C. Greubel_, Sep 07 2023

%Y Cf. A001482 - A001488, A001490, A047265, A047638 - A047649, A047655, A341243.

%K sign

%O 2,2

%A _N. J. A. Sloane_

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