A047654 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^2 in powers of x.

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, -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, 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

Offset: 2

Author

N. J. A. Sloane

Keywords

• sign

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..10000
• H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)
• H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.

Crossrefs

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

Programs

Magma

m:=120;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^2 )); // G. C. Greubel, Sep 07 2023

Maple

g:= proc(n) option remember; `if`(n=0, 1, add(add([-d, d, -2*d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0, g(n)),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 2):
seq(a(n), n=2..94);  # Alois P. Heinz, Feb 07 2021

Mathematica

nmax=94; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] - 1)^2, {x, 0, nmax}], x]//Drop[#, 2] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=2}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x,0, 125}], x], k]] (* G. C. Greubel, Sep 07 2023 *)

PARI

seq(n)={Vec((prod(j=1, n, 1-(-x)^j + O(x^n)) - 1)^2)} \\ Andrew Howroyd, Feb 07 2021

SageMath

from sage.modular.etaproducts import qexp_eta
m=125; k=2;
def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047654_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A047654_list(m); a[k:] # G. C. Greubel, Sep 07 2023

Formula

a(n) = [x^n]( QPochhammer(-x) - 1 )^2. - G. C. Greubel, Sep 07 2023

Extensions

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021