A047647 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^22 in powers of x.

1, -22, 231, -1540, 7293, -25872, 69971, -140822, 183711, -25102, -634480, 2027804, -3817814, 4439116, -919600, -9829270, 27660479, -44779042, 43632974, -1898820, -92518261, 219961214, -313463842, 267448104, 15757973, -547042056, 1173033400

Offset

22,2

Links

Alois P. Heinz, Table of n, a(n) for n = 22..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 )^22. - G. C. Greubel, Sep 05 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, 22):
seq(a(n), n=22..48);  # Alois P. Heinz, Feb 07 2021

Mathematica

nmax=48; CoefficientList[Series[(Product[(1-(-x)^j), {j, nmax}] - 1)^22, {x, 0, nmax}], x]//Drop[#, 22] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=22}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x, 0, 75}], x], k]] (* G. C. Greubel, Sep 05 2023 *)

Prog

(Magma)
m:=75;
R<x>:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(22) )); // G. C. Greubel, Sep 05 2023
(SageMath)
from sage.modular.etaproducts import qexp_eta
m=75; k=22;
def f(k, x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047647_list(prec):
    P.<x> = PowerSeriesRing(QQ, prec)
    return P( f(k, x) ).list()
a=A047647_list(m); a[k:] # G. C. Greubel, Sep 05 2023
(PARI) my(N=55, x='x+O('x^N)); Vec((eta(-x)-1)^22) \\ Michel Marcus, Sep 05 2023

Crossrefs

Cf. A001482 - A001488, A001490, A047638 - A047646, A047648, A047649, A047654, A047655, A341243.

Sequence in context: A133719 A008948 A003909 * A010938 A022617 A082205

Adjacent sequences: A047644 A047645 A047646 * A047648 A047649 A047650

Keyword

sign

Author

N. J. A. Sloane

Extensions

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

Status

approved