A019581 Place n distinguishable balls in n boxes (in n^n ways); let f(n,k) = number of ways that max in any box is k, for 1<=k<=n; sequence gives f(n,2).

0, 2, 18, 180, 2100, 28800, 458640, 8361360, 172141200, 3954484800, 100330876800, 2786980996800, 84133667217600, 2742770705875200, 96032990237184000, 3594185336405664000, 143193231131382432000, 6050494745192177280000, 270263142944131873536000

Offset: 1

Author

Lee Corbin (lcorbin(AT)tsoft.com), N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A019576.

Column k=2 of A019575. - Alois P. Heinz, Jul 29 2014

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-j, i-1)/j!, j=0..min(2, n))))
    end:
a:= n-> n! *(b(n$2) -1):
seq(a(n), n=1..30);  # Alois P. Heinz, Jul 29 2014

Mathematica

a[n_] := n! (Hypergeometric2F1[1/2 - n/2, -n/2, 1, 2] - 1); Array[a, 30] (* Jean-François Alcover, Feb 18 2016 *)

PARI

a(n) = sum(d=1, n\2, n!^2 / (2^d * (n-2*d)! * d!^2)); \\ Michel Marcus, Aug 13 2013

Formula

a(n) = sum(d=1..floor(n/2), n!^2 / ( 2^d * (n-2*d)! * d! * d! ) ).