A281881 Triangle read by rows: T(n,k) (n>=1, 2<=k<=n+1) is the number of k-sequences of balls colored with at most n colors such that exactly one ball is of a color seen previously in the sequence.

1, 2, 6, 3, 18, 36, 4, 36, 144, 240, 5, 60, 360, 1200, 1800, 6, 90, 720, 3600, 10800, 15120, 7, 126, 1260, 8400, 37800, 105840, 141120, 8, 168, 2016, 16800, 100800, 423360, 1128960, 1451520

Offset: 1

Author

Jeremy Dover, Feb 01 2017

Keywords

• nonn
• tabl

Comments

Number of k-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 -> T(1,2) = 1.
n=2 => AA, BB -> T(2,2) = 2; AAB, ABA, BAA, BBA, BAB, ABB -> T(2,3) = 6.
Triangle starts:
   1
   2,   6
   3,  18,   36
   4,  36,  144,   240
   5,  60,  360,  1200,   1800
   6,  90,  720,  3600,  10800,   15120
   7, 126, 1260,  8400,  37800,  105840,   141120
   8, 168, 2016, 16800, 100800,  423360,  1128960,  1451520
   9, 216, 3024, 30240, 226800, 1270080,  5080320, 13063680,  16329600
  10, 270, 4320, 50400, 453600, 3175200, 16934400, 65318400, 163296000, 199584000
		

Links

• Jeremy Dover, Table of n, a(n) for n = 1..999
• Jeremy Dover, Answer to Cumulative Distribution Function of Collision Counts

Crossrefs

Columns of table:

T(n,2) = A000027(n)

T(n,3) = A028896(n)

Other sequences in table:

T(n,n+1) = A001286(n)

T(n,n) = A001804(n), n>=2

Programs

Mathematica

Table[Binomial[k, 2] n!/(n + 1 - k)!, {n, 8}, {k, 2, n + 1}] // Flatten (* Michael De Vlieger, Feb 02 2017 *)

Formula

T(n,k) = binomial(k,2)*n!/(n+1-k)!.

T(n,k) = n*T(n-1,k-1) + (k-1)*n!/(n+1-k)!.