%I
%S 1,2,8,3,24,60,4,48,240,480,5,80,600,2400,4200,6,120,1200,7200,25200,
%T 40320,7,168,2100,16800,88200,282240,423360,8,224,3360,33600,235200,
%U 1128960,3386880,4838400,9,288,5040,60480,529200,3386880,15240960,43545600,59875200,10,360,7200,100800,1058400,8467200,50803200,217728000,598752000,798336000
%N Triangle read by rows: T(n,k) (n>=1, 3<=k<=n+2) is the number of ksequences of balls colored with at most n colors such that exactly three balls are the same color as some other ball in the sequence
%C Note that the three matching balls are necessarily the same color.
%F T(n, k) = binomial(k,3)*n!/(n+2k)!.
%e n=1 => AAA > T(1,3)=1;
%e n=2 => AAA,BBB > T(2,3)=2;
%e AAAB,AABA,ABAA,BAAA,BBBA,BBAB,BABB,ABBB > T(2,4)=8.
%e Triangle begins:
%e 1;
%e 2, 8;
%e 3, 24, 60;
%e 4, 48, 240, 480;
%e 5, 80, 600, 2400, 4200;
%e ...
%o (PARI) T(n, k) = binomial(k,3)*n!/(n+2k)!;
%o tabl(nn) = for (n=1, nn, for (k=3, n+2, print1(T(n,k), ", ")); print()); \\ _Michel Marcus_, Sep 29 2017
%Y Columns of table: T(n,3) = A000027(n), T(n,4) = A033996(n).
%Y Other sequences in table: T(n,n+2) = A005990(n+1).
%K nonn,tabl
%O 1,2
%A _Jeremy Dover_, Sep 26 2017
