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

1,2

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

Links

Jeremy Dover, Table of n, a(n) for n = 1..999

Jeremy Dover, Answer to Cumulative Distribution Function of Collision Counts

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)!.

Example

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

Mathematica

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

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

Sequence in context: A379559 A050125 A178667 * A377876 A206493 A359256

Adjacent sequences: A281878 A281879 A281880 * A281882 A281883 A281884

Keyword

nonn,tabl

Author

Jeremy Dover, Feb 01 2017

Status

approved