A306153 Inverse Weigh transform of (-1)^n * n!.

-1, 1, -4, 18, -92, 572, -4156, 34177, -314368, 3199844, -35703996, 433422067, -5687955724, 80256874912, -1211781887796, 19496946568897, -333041104402860, 6019770246910128, -114794574818830716, 2303332661419442477, -48509766592884311132, 1069983257387132347080

Offset

1,3

Links

Alois P. Heinz, Table of n, a(n) for n = 1..449

N. J. A. Sloane, Transforms

Formula

a(n) ~ (-1)^n * n! * (1 - 1/n - 1/n^2 - 4/n^3 - 23/n^4 - 171/n^5 - 1542/n^6 - 16241/n^7 - 194973/n^8 - 2622610/n^9 - 39027573/n^10 - ...), for coefficients see A113869. - Vaclav Kotesovec, Nov 27 2020

Example

(1+x)^(-1)*(1+x^2)*(1+x^3)^(-4)*(1+x^4)^18* ... = 1 - x + 2*x^2 - 6*x^3 + 24*x^4 - ... .

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> (-1)^n*n!-b(n, n-1):
seq(a(n), n=1..24);  # Alois P. Heinz, Jun 23 2018

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[a[i], j] b[n - i j, i - 1], {j, 0, n/i}]]];
a[n_] := (-1)^n n! - b[n, n - 1] // FullSimplify;
Array[a, 24] (* Jean-François Alcover, Nov 27 2020, after Alois P. Heinz *)

Crossrefs

Cf. A168246.

Sequence in context: A337010 A081923 A020064 * A123589 A245103 A200717

Adjacent sequences: A306150 A306151 A306152 * A306154 A306155 A306156

Keyword

sign

Author

Seiichi Manyama, Jun 23 2018

Status

approved