A047645 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^20 in powers of x.

1, -20, 190, -1140, 4825, -15124, 35320, -57760, 45220, 80560, -405954, 910460, -1289340, 852340, 1259530, -5357924, 10151510, -12048660, 5883350, 12186960, -40135713, 66244280, -69648870, 28191460, 66920755, -195366168, 300881530

Offset

20,2

Links

Alois P. Heinz, Table of n, a(n) for n = 20..10000

H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)

H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.

Formula

a(n) = [x^n]( QPochhammer(-x) - 1 )^20. - G. C. Greubel, Sep 06 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, 20):
seq(a(n), n=20..46);  # Alois P. Heinz, Feb 07 2021

Mathematica

nmax=46; CoefficientList[Series[(Product[(1-(-x)^j), {j, nmax}] -1)^20, {x, 0, nmax}], x]//Drop[#, 20] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=20}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x, 0, 75}], x], k]] (* G. C. Greubel, Sep 06 2023 *)

Prog

(Magma)
m:=80;
R<x>:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(20) )); // G. C. Greubel, Sep 06 2023
(SageMath)
from sage.modular.etaproducts import qexp_eta
m=75; k=20;
def f(k, x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047645_list(prec):
    P.<x> = PowerSeriesRing(QQ, prec)
    return P( f(k, x) ).list()
a=A047645_list(m); a[k:] # G. C. Greubel, Sep 06 2023
(PARI) my(N=44, x='x+O('x^N)); Vec((eta(-x)-1)^20) \\ Joerg Arndt, Sep 06 2023

Crossrefs

Cf. A001482 - A001488, A001490, A047638 - A047644, A047646 - A047649, A047654, A047655, A341243.

Sequence in context: A211153 A210429 A367780 * A010936 A014806 A022615

Adjacent sequences: A047642 A047643 A047644 * A047646 A047647 A047648

Keyword

sign

Author

N. J. A. Sloane

Extensions

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

Status

approved