A001485 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^7 in powers of x.

1, -7, 21, -35, 28, 21, -105, 181, -189, 77, 140, -385, 546, -511, 252, 203, -693, 1029, -1092, 798, -203, -581, 1281, -1708, 1687, -1232, 413, 602, -1485, 2233, -2366, 2009, -1099, 14, 1099, -2072, 2667, -2807, 2254, -1477, 0, 1057, -2346, 2744, -3017, 2457

Offset: 7

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 = 7..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, A001484, A001486, 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)^7 )); // 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, 7):
seq(a(n), n=7..52);  # Alois P. Heinz, Feb 07 2021

Mathematica

nmax = 52; CoefficientList[Series[(Product[(1 - (-x)^j), {j, 1, nmax}] - 1)^7, {x, 0, nmax}], x] // Drop[#, 7] & (* Ilya Gutkovskiy, Feb 07 2021 *)
Drop[CoefficientList[Series[(QPochhammer[-x] -1)^7, {x,0,102}], x], 7] (* G. C. Greubel, Sep 04 2023 *)

PARI

my(N=70,x='x+O('x^N)); Vec((eta(-x)-1)^7) \\ Joerg Arndt, Sep 04 2023

SageMath

m=100; k=7;
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 A001485_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A001485_list(m); a[k:] # G. C. Greubel, Sep 04 2023

Formula

a(n) = [x^n] ( QPochhammer(-x) - 1 )^7. - G. C. Greubel, Sep 04 2023

Extensions

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021