A281944 Triangle read by rows: T(n,k) (n>=1, 3<=k<=n+2) is the number of k-sequences of balls colored with n colors such that exactly two balls are of a color seen previously in the sequence.

1, 2, 14, 3, 42, 150, 4, 84, 600, 1560, 5, 140, 1500, 7800, 16800, 6, 210, 3000, 23400, 100800, 191520, 7, 294, 5250, 54600, 352800, 1340640, 2328480, 8, 392, 8400, 109200, 940800, 5362560, 18627840, 30240000, 9, 504, 12600, 196560, 2116800, 16087680, 83825280, 272160000, 419126400, 10, 630, 18000, 327600, 4233600, 40219200, 279417600, 1360800000, 4191264000, 6187104000

Offset: 1

Author

Jeremy Dover, Feb 02 2017

Keywords

• nonn
• tabl

Examples

			n=1 => AAA -> T(1,3)=1
n=2 => AAA,BBB -> T(2,3)=2
   AAAB,AABA,ABAA,BAAA,BBBA,BBAB,BABB,ABBB,AABB,ABAB,ABBA,BAAB,BABA,BBAA -> T(2,4)=14
Triangle starts:
   1
   2,  14
   3,  42,   150
   4,  84,   600,   1560
   5, 140,  1500,   7800,   16800
   6, 210,  3000,  23400,  100800,   191520
   7, 294,  5250,  54600,  352800,  1340640,  2328480
   8, 392,  8400, 109200,  940800,  5362560, 18627840,  30240000
   9, 504, 12600, 196560, 2116800, 16087680, 83825280, 272160000, 419126400
		

Links

• Jeremy Dover, Table of n, a(n) for n = 1..1035
• Jeremy Dover, Answer to Cumulative Distribution Function of Collision Counts

Crossrefs

Columns of table: T(n,3) = A000027(n), T(n,4) = A163756(n).

Other sequences in table: T(n,n+2) = A037960(n).

Programs

Mathematica

Table[(Binomial[k, 3] + 3 Binomial[k, 4]) n!/(n + 2 - k)!, {n, 12}, {k, 3, n + 2}] // Flatten (* Michael De Vlieger, Feb 05 2017 *)

PARI

T(n, k) = (binomial(k,3) + 3*binomial(k,4)) * n! / (n+2-k)!;
tabl(nn) = for (n=1, nn, for (k=3, n+2, print1(T(n,k), ", ")); print()); \\ Michel Marcus, Feb 04 2017

Formula

T(n, k) = (binomial(k,3) + 3*binomial(k,4)) * n! / (n+2-k)!.

T(n, k) = n*T(n-1,k-1) + (k-2)*A281881(n,k-1).