A281912 Number of sequences of balls colored with at most n colors such that exactly one ball is of a color seen earlier in the sequence.

1, 8, 57, 424, 3425, 30336, 294553, 3123632, 36003969, 448816600, 6022033721, 86587079448, 1328753602657, 21683227579664, 375013198304025, 6853321766162656, 131976208783240193, 2671430511854158632, 56709161712552286009, 1259836187316759240200

Offset: 1

Author

Jeremy Dover, Feb 01 2017

Keywords

• nonn

Comments

Note that any such sequence has at least 2 balls, and at most n+1

Number of sequences of balls colored with at most n colors such that exactly two balls are the same color as some other ball in the sequence (necessarily each other). - Jeremy Dover, Sep 26 2017

Examples

			n=1 => AA -> a(1) = 1.
n=2 => AA,BB,AAB,ABA,BAA,BBA,BAB,ABB -> a(2) = 8.
		

Links

• Jeremy Dover, Table of n, a(n) for n = 1..99

Crossrefs

Cf. A093964.

Row sums of triangle A281881. - Jeremy Dover, Sep 26 2017

Programs

Maple

a:= proc(n) option remember;
      `if`(n<2, 1, a(n-1)*(n+2)/(n-1)-a(n-2))*n
    end:
seq(a(n), n=1..25);  # Alois P. Heinz, Feb 02 2017

Mathematica

Table[n!*Sum[Binomial[k, 2]/(n + 1 - k)!, {k, 2, n + 1}], {n, 20}] (* Michael De Vlieger, Feb 02 2017 *)

Formula

a(n) = n! * Sum_{k=2..n+1} binomial(k,2)/(n+1-k)!.

a(n) = n if n < 2, a(n) = n*((n+2)/(n-1)*a(n-1) - a(n-2)) for n >= 2. - Alois P. Heinz, Feb 02 2017

a(n)/n! ~ e*n^2/2. - Vaclav Kotesovec, Feb 03 2017