A292930 Triangle read by rows: T(n,k) (n>=1, 3<=k<=n+2) is the number of k-sequences of balls colored with at most n colors such that exactly three balls are the same color as some other ball in the sequence

1, 2, 8, 3, 24, 60, 4, 48, 240, 480, 5, 80, 600, 2400, 4200, 6, 120, 1200, 7200, 25200, 40320, 7, 168, 2100, 16800, 88200, 282240, 423360, 8, 224, 3360, 33600, 235200, 1128960, 3386880, 4838400, 9, 288, 5040, 60480, 529200, 3386880, 15240960, 43545600, 59875200, 10, 360, 7200, 100800, 1058400, 8467200, 50803200, 217728000, 598752000, 798336000

Offset

1,2

Comments

Note that the three matching balls are necessarily the same color.

Links

Table of n, a(n) for n=1..55.

Formula

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

Example

n=1 => AAA -> T(1,3)=1;
n=2 => AAA,BBB -> T(2,3)=2;
       AAAB,AABA,ABAA,BAAA,BBBA,BBAB,BABB,ABBB -> T(2,4)=8.
Triangle begins:
  1;
  2, 8;
  3, 24, 60;
  4, 48, 240, 480;
  5, 80, 600, 2400, 4200;
  ...

Prog

(PARI) T(n, k) = binomial(k, 3)*n!/(n+2-k)!;
tabl(nn) = for (n=1, nn, for (k=3, n+2, print1(T(n, k), ", ")); print()); \\ Michel Marcus, Sep 29 2017

Crossrefs

Columns of table: T(n,3) = A000027(n), T(n,4) = A033996(n).

Other sequences in table: T(n,n+2) = A005990(n+1).

Sequence in context: A153188 A193976 A264244 * A126951 A282637 A256411

Adjacent sequences: A292927 A292928 A292929 * A292931 A292932 A292933

Keyword

nonn,tabl

Author

Jeremy Dover, Sep 26 2017

Status

approved