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

23,2

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.

Formula

a(n) = [x^n]( QPochhammer(-x) - 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 *)

Prog

(Magma)
m:=75;
R<x>:=PowerSeriesRing(Integers(), m);
Coefficients(R!( ((&*[1-(-x)^j: j in [1..m+2]]) -1)^(23) )); // G. C. Greubel, 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.<x> = PowerSeriesRing(QQ, prec)
    return P( f(k, x) ).list()
a=A047648_list(m); a[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

Crossrefs

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

Sequence in context: A003911 A061977 A034777 * A010939 A022618 A042018

Adjacent sequences: A047645 A047646 A047647 * A047649 A047650 A047651

Keyword

sign

Author

N. J. A. Sloane

Extensions

Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021

Status

approved