A001484 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^6 in powers of x.

1, -6, 15, -20, 9, 24, -65, 90, -75, 6, 90, -180, 220, -180, 66, 110, -264, 360, -365, 264, -66, -178, 375, -510, 496, -414, 180, 60, -330, 570, -622, 582, -390, 220, 96, -300, 621, -630, 705, -492, 300, 0, -235, 420, -570, 594, -735, 420, -420, -120, 219, -586, 360

Offset: 6

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

• Robert Israel, Table of n, a(n) for n = 6..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, A001483, A001485 - 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)^6 )); // G. C. Greubel, Sep 04 2023

Maple

N:= 100:
S:= series((mul(1-(-x)^j,j=1..N)-1)^6,x,N+1):
seq(coeff(S,x,j),j=6..N); # Robert Israel, Feb 05 2019

Mathematica

Drop[CoefficientList[Series[(QPochhammer[-x] -1)^6, {x,0,102}], x], 6] (* G. C. Greubel, Sep 04 2023 *)

PARI

my(N=70,x='x+O('x^N)); Vec((eta(-x)-1)^6) \\ Joerg Arndt, Sep 04 2023

SageMath

m=100; k=6;
def f(k,x): return (-1 + product( (1+x^j)*(1-x^(2*j))/(1+x^(2*j)) for j in range(1,m+2) ) )^k
def A001484_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A001484_list(m); a[k:] # G. C. Greubel, Sep 04 2023

Formula

a(n) = [x^n] ( QPochhammer(-x) - 1 )^6. - G. C. Greubel, Sep 04 2023

Extensions

Edited by Robert Israel, Feb 05 2019