A001484 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^6 in powers of x. (Formerly M4107 N1704)

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

6,2

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)

Formula

a(n) = [x^n] ( QPochhammer(-x) - 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 *)

Prog

(Magma)
m:=102;
R<x>:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^6 )); // G. C. Greubel, 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.<x> = PowerSeriesRing(QQ, prec)
    return P( f(k, x) ).list()
a=A001484_list(m); a[k:] # 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

Crossrefs

Cf. A001482, A001483, A001485 - A001488, A047638 - A047649, A047654, A047655, A341243.

Sequence in context: A044058 A105139 A002599 * A217480 A335417 A173680

Adjacent sequences: A001481 A001482 A001483 * A001485 A001486 A001487

Keyword

sign

Author

N. J. A. Sloane

Extensions

Edited by Robert Israel, Feb 05 2019

Status

approved