A047655 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^3 in powers of x.

1, -3, 3, -1, -3, 6, -6, 6, 0, -3, 6, -9, 8, -6, 0, 0, -6, 6, -13, 3, -6, 3, 0, -3, 6, -9, 6, -3, 6, 0, 6, 6, -3, 11, 0, 6, 0, 9, 0, 0, 0, -3, 13, 0, 0, -6, 0, -6, 3, -3, -6, 0, -15, -6, -3, 0, -6, 0, -6, 0, -6, -6, 0, -11, 0, 0, -6, 0, 6, 0, 6, 0, 0, 0, -3, 19, 12, -3, 0, 0, 6, 6, 6, 6, 0, 0, 6, 0, 21, 3

Offset: 3

Author

N. J. A. Sloane

Keywords

• sign

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..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, A047654, A341243.

Programs

Magma

m:=120;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^3 )); // 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, 3):
seq(a(n), n=3..92);  # Alois P. Heinz, Feb 07 2021

Mathematica

nmax=92; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] - 1)^3, {x,0,nmax}], x]//Drop[#, 3] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=3}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x,0, 125}], x], k]] (* G. C. Greubel, Sep 07 2023 *)

PARI

my(x='x+O('x^99)); Vec((eta(-x)-1)^3) \\ Joerg Arndt, Sep 07 2023

SageMath

from sage.modular.etaproducts import qexp_eta
m=125; k=3;
def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047655_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A047655_list(m); a[k:] # G. C. Greubel, Sep 07 2023

Formula

a(n) = [x^n]( QPochhammer(-x) - 1 )^3. - G. C. Greubel, Sep 07 2023

Extensions

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021