A001482 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^4 in powers of x.

1, -4, 6, -4, -3, 12, -16, 16, -6, -8, 18, -28, 26, -20, 2, 12, -23, 32, -36, 28, -6, 4, 22, -20, 39, -32, 32, -12, 2, 16, -12, 24, -40, 28, -34, 0, -6, -16, 0, -40, 6, -36, 26, -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

Offset: 4

Author

N. J. A. Sloane

Keywords

• sign

References

• H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
• 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 = 4..10000
• H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)

Crossrefs

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

Programs

Magma

m:=102;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( (1 - (&*[1-(-x)^j: j in [1..m+2]]))^4 )); // 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, 4):
seq(a(n), n=4..73);  # Alois P. Heinz, Feb 07 2021

Mathematica

nmax = 73; CoefficientList[Series[(Product[(1 - (-x)^j), {j, 1, nmax}] - 1)^4, {x, 0, nmax}], x] // Drop[#, 4] & (* Ilya Gutkovskiy, Feb 07 2021 *)
Drop[CoefficientList[Series[(1 -QPochhammer[-x])^4, {x,0,100}], x], 4] (* G. C. Greubel, Sep 04 2023 *)

PARI

my(N=70,x='x+O('x^N)); Vec((eta(-x)-1)^4) \\ Joerg Arndt, Sep 04 2023

SageMath

m=100
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
def A001482_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f4(x) ).list()
a=A001482_list(m); a[4:] # G. C. Greubel, Sep 04 2023

Extensions

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021