A292999 Triangle read by rows: T(n,k) (n >= 1, 4 <= k <= n+3) is the number of k-sequences of balls colored with at most n colors such that exactly four balls are the same color as some other ball in the sequence.

1, 8, 10, 21, 120, 90, 40, 420, 1440, 840, 65, 1000, 6300, 16800, 8400, 96, 1950, 18000, 88200, 201600, 90720, 133, 3360, 40950, 294000, 1234800, 2540160, 1058400, 176, 5320, 80640, 764400, 4704000, 17781120, 33868800, 13305600, 225, 7920, 143640, 1693440, 13759200, 76204800, 266716800, 479001600, 179625600

Offset

1,2

Links

Table of n, a(n) for n=1..45.

Formula

a(n) = binomial(k,4)*n!*(1/(n+3-k)! + 3/(n+2-k)!) (with the convention that 3/(-1)! = 0 when k=n+3).

Example

For n=1: AAAA -> T(1,4)=1.
For n=2: AAAA,BBBB,AABB,ABAB,ABBA,BAAB,BABA,BBAA -> T(2,4)=8; AAAAB,AAABA,AABAA,ABAAA,BAAAA,BBBBA,BBBAB,BBABB,BABBB,ABBBB -> T(2,5)=10.
Triangle starts:
    1;
    8,   10;
   21,  120,     90;
   40,  420,   1440,     840;
   65, 1000,   6300,   16800,     8400;
   96, 1950,  18000,   88200,   201600,    90720;
  133, 3360,  40950,  294000,  1234800,  2540160,   1058400;
  176, 5320,  80640,  764400,  4704000, 17781120,  33868800, 13305600;
  225, 7920, 143640, 1693440, 13759200, 76204800, 266716800, ... .

Mathematica

Table[Binomial[k, 4] n! (1/(n + 3 - k)! + 3/(n + 2 - k)!), {n, 9}, {k, 4, n + 3}] // Flatten (* Michael De Vlieger, Sep 30 2017 *)

Crossrefs

Columns of the table: T(n,4) = A000567(n), T(n,5) = 10*A007586(n-1), T(n,6) = 90*A220212(n-2).

Diagonals of the table: T(n,n+3) = A061206(n), T(n+1,n+3) = 8*A005461(n), T(n-1,n) = 21*A001755(n), T(n,n) = 40*A001811(n), T(n,n-1) = 65*A001777(n), T(n+6,n+4) = A062194(n).

Sequence in context: A073619 A338820 A302429 * A216047 A032488 A102844

Adjacent sequences: A292996 A292997 A292998 * A293000 A293001 A293002

Keyword

nonn,tabl

Author

Jeremy Dover, Sep 27 2017

Status

approved