A292541 a(n) is n-th term of the Euler transform of -n,1,1,1,... .

1, -1, 2, -3, 5, -9, 18, -39, 88, -200, 449, -988, 2131, -4527, 9540, -20090, 42510, -90596, 194299, -418105, 899493, -1929000, 4116944, -8742002, 18484225, -38974978, 82086786, -172927251, 364700265, -770223900, 1628602725, -3445907334, 7291399538

Offset: 0

Author

Alois P. Heinz, Sep 18 2017

Keywords

• sign

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..3260

Crossrefs

Cf. A292463.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0 or i<2,
      binomial(k+n-1, n), add(b(n-i*j, i-1, k), j=0..n/i))
    end:
a:= n-> b(n$2, -n):
seq(a(n), n=0..35);
# second Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1, add(
      (numtheory[sigma](j)+k-1)*b(n-j, k), j=1..n)/n)
    end:
a:= n-> b(n, -n):
seq(a(n), n=0..35);
# third Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1, `if`(k=1,
      combinat[numbpart](n), b(n, k+1)-b(n-1, k+1)))
    end:
a:= n-> b(n, -n):
seq(a(n), n=0..35);

Mathematica

Table[SeriesCoefficient[(1 - x)^n*Product[1/(1 - x^k), {k, 2, n}], {x, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, May 07 2018 *)
b[n_, k_] := b[n, k] = If[n == 0, 1, If[k == 1, PartitionsP[n], b[n, k + 1] - b[n - 1, k + 1]]]; Table[b[n, -n], {n, 0, 40}] (* Vaclav Kotesovec, May 07 2018, after Alois P. Heinz *)

Formula

a(n) = [x^n] (1-x)^n / Product_{j=2..n} (1-x^j).

a(n) ~ (-1)^n * exp(Pi*sqrt(n/3)/2 + Pi^2/96) * 2^(n - 1/2) / (sqrt(3)*n). - Vaclav Kotesovec, May 07 2018