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
KEYWORD
sign
AUTHOR
EXTENSIONS
Definition and offset edited by Ilya Gutkovskiy, Feb 07 2021
STATUS
approved