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

Author

Jeremy Dover, Sep 27 2017

Keywords

• nonn
• tabl

Examples

			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, ... .
		

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).

Programs

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 *)

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).