A001488 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^10 in powers of x.

1, -10, 45, -120, 200, -162, -160, 810, -1530, 1730, -749, -1630, 4755, -7070, 6700, -2450, -5295, 14070, -20010, 19350, -10157, -6290, 25515, -40660, 44940, -34268, 9180, 24510, -57195, 78060, -79087, 56610, -13935, -39600, 89805, -121638, 125405

Offset: 10

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 = 10..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 - A001487, A001490, A047638 - A047649, A047654, A047655, A341243.

Programs

Magma

m:=102;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^10 )); // 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, 10):
seq(a(n), n=10..46);  # Alois P. Heinz, Feb 07 2021

Mathematica

nmax=46; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] -1)^10, {x,0,nmax}], x]//Drop[#,10] & (* Ilya Gutkovskiy, Feb 07 2021 *)
Drop[CoefficientList[Series[(QPochhammer[-x] -1)^10, {x,0,102}], x],10] (* G. C. Greubel, Sep 04 2023 *)

PARI

my(N=55,x='x+O('x^N)); Vec((eta(-x)-1)^10) \\ Joerg Arndt, Sep 05 2023

SageMath

from sage.modular.etaproducts import qexp_eta
m=100; k=10;
def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A001488_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A001488_list(m); a[k:] # G. C. Greubel, Sep 04 2023

Formula

a(n) = [x^n]( QPochhammer(-x) - 1 )^10. - G. C. Greubel, Sep 04 2023

Extensions

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021