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

1,2

Links

Jeremy Dover, Table of n, a(n) for n = 1..1035

Jeremy Dover, Answer to Cumulative Distribution Function of Collision Counts

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).

Example

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

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 *)

Prog

(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

Crossrefs

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

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

Sequence in context: A138907 A336837 A276189 * A306724 A316909 A103979

Adjacent sequences: A281941 A281942 A281943 * A281945 A281946 A281947

Keyword

nonn,tabl

Author

Jeremy Dover, Feb 02 2017

Status

approved