A047648 Expansion of (Product_{j>=1} (1-(-x)^j) - 1)^23 in powers of x.

1, -23, 253, -1771, 8832, -33143, 95611, -209231, 317009, -181401, -686642, 2828977, -6099278, 8422623, -4906406, -10919687, 41968146, -78977952, 93297545, -40351223, -117265247, 367581446, -606562624, 631382751, -207879980, -777907725, 2132043121

Offset: 23

Author

N. J. A. Sloane

Keywords

• sign

Links

• Alois P. Heinz, Table of n, a(n) for n = 23..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.

Crossrefs

Cf. A001482 - A001488, A001490, A047638 - A047647, A047649, A047654, A047655, A341243.

Programs

Magma

m:=75;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(23) )); // 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, 23):
seq(a(n), n=23..49);  # Alois P. Heinz, Feb 07 2021

Mathematica

nmax=49; CoefficientList[Series[(Product[(1-(-x)^j), {j,nmax}] -1)^23, {x,0,nmax}], x]//Drop[#, 23] & (* Ilya Gutkovskiy, Feb 07 2021 *)
With[{k=23}, Drop[CoefficientList[Series[(QPochhammer[-x] -1)^k, {x,0, 75}], x], k]] (* G. C. Greubel, Sep 05 2023 *)

PARI

my(N=44, x='x+O('x^N)); Vec((eta(-x)-1)^23) \\ Joerg Arndt, Sep 05 2023

SageMath

from sage.modular.etaproducts import qexp_eta
m=75; k=23;
def f(k,x): return (-1 + qexp_eta(QQ[['q']], m+2).subs(q=-x) )^k
def A047648_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(k,x) ).list()
a=A047648_list(m); a[k:] # G. C. Greubel, Sep 05 2023

Formula

a(n) = [x^n]( QPochhammer(-x) - 1 )^23. - G. C. Greubel, Sep 05 2023

Extensions

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021