A168246 Inverse Weigh transform of n!.

1, 2, 4, 19, 92, 576, 4156, 34178, 314368, 3199936, 35703996, 433422071, 5687955724, 80256879068, 1211781887796, 19496946568898, 333041104402860, 6019770247224496, 114794574818830716, 2303332661419442569, 48509766592884311132, 1069983257387168051076

Offset: 1

Author

Vladeta Jovovic, Nov 21 2009

Keywords

• easy
• nonn

Links

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

Crossrefs

Cf. A000142, A112354, A261052 (Weigh transform of n!).

Programs

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:= proc(n) option remember; n! -b(n, n-1) end:
seq(a(n), n=1..30);  # Alois P. Heinz, Jun 11 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_] := a[n] = n! - b[n, n - 1];
Array[a, 30] (* Jean-François Alcover, Sep 16 2019, after Alois P. Heinz *)

PARI

seq(n)={dirdiv(Vec(log(1+x*Ser(vector(n, n, n!)))), -vector(n, n, (-1)^n/n))} \\ Andrew Howroyd, Jun 22 2018

Formula

Product_{k>=1} (1+x^k)^a(k) = Sum_{n>=0} n! x^n.

a(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