A047641 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^16 in powers of x.

1, -16, 120, -560, 1804, -4128, 6312, -3920, -10530, 42208, -82752, 99584, -39460, -141200, 422568, -673936, 660941, -144720, -938840, 2301568, -3257188, 2916592, -628040, -3492160, 8217536, -11341568, 10408280, -3885040, -7668720, 21033408

Offset

16,2

Links

Alois P. Heinz, Table of n, a(n) for n = 16..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 )^16. - G. C. Greubel, Sep 07 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, 16):
seq(a(n), n=16..45);  # Alois P. Heinz, Feb 07 2021

Mathematica

nmax=45; CoefficientList[Series[(Product[(1-(-x)^j), {j, nmax}] -1)^16, {x, 0, nmax}], x]//Drop[#, 16] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=16}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x, 0, 75}], x], k]] (* G. C. Greubel, Sep 07 2023 *)

Prog

(Magma)
m:=80;
R<x>:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(16) )); // G. C. Greubel, Sep 07 2023
(SageMath)
from sage.modular.etaproducts import qexp_eta
m=75; k=16;
def f(k, x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047641_list(prec):
    P.<x> = PowerSeriesRing(QQ, prec)
    return P( f(k, x) ).list()
a=A047641_list(m); a[k:] # G. C. Greubel, Sep 07 2023
(PARI) my(x='x+O('x^40)); Vec((eta(-x)-1)^16) \\ Joerg Arndt, Sep 07 2023

Crossrefs

Cf. A001482 - A001488, A001490, A047638 - A047640, A047642 - A047649, A047654, A047655, A341243.

Sequence in context: A190049 A317227 A138571 * A010932 A014805 A022611

Adjacent sequences: A047638 A047639 A047640 * A047642 A047643 A047644

Keyword

sign

Author

N. J. A. Sloane

Extensions

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

Status

approved