A001486 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^8 in powers of x.

1, -8, 28, -56, 62, 0, -148, 328, -419, 280, 140, -728, 1232, -1336, 848, 224, -1582, 2688, -3072, 2408, -742, -1568, 3836, -5264, 5306, -3744, 924, 2576, -5686, 7792, -8092, 6272, -2751, -1848, 6008, -9296, 10556, -9800, 6692, -2240, -3206, 8168, -11524

Offset: 8

Author

N. J. A. Sloane

Keywords

• sign

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Alois P. Heinz, Table of n, a(n) for n = 8..10000
• H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
• H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)

Crossrefs

Cf. A001482 - A001485, A001487, A001488, A047638 - A047649, A047654, A047655, A341243.

Programs

Magma

m:=102;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^8 )); // G. C. Greubel, Sep 04 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, 8):
seq(a(n), n=8..50);  # Alois P. Heinz, Feb 07 2021

Mathematica

nmax=50; CoefficientList[Series[(Product[(1 -(-x)^j), {j,nmax}] -1)^8, {x,0,nmax}], x]//Drop[#,8] & (* Ilya Gutkovskiy, Feb 07 2021 *)
Drop[CoefficientList[Series[(QPochhammer[-x] -1)^8, {x,0,102}], x], 8] (* G. C. Greubel, Sep 04 2023 *)

PARI

my(N=55,x='x+O('x^N)); Vec((eta(-x)-1)^8) \\ Joerg Arndt, Sep 05 2023

SageMath

from sage.modular.etaproducts import qexp_eta
m=100; k=8;
def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A001486_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A001486_list(m); a[k:] # G. C. Greubel, Sep 04 2023

Formula

a(n) = [x^n]( QPochhammer(-x) - 1 )^8. - G. C. Greubel, Sep 04 2023

Extensions

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021