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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.

%I A292999 #23 Oct 18 2017 21:56:50
%S A292999 1,8,10,21,120,90,40,420,1440,840,65,1000,6300,16800,8400,96,1950,
%T A292999 18000,88200,201600,90720,133,3360,40950,294000,1234800,2540160,
%U A292999 1058400,176,5320,80640,764400,4704000,17781120,33868800,13305600,225,7920,143640,1693440,13759200,76204800,266716800,479001600,179625600
%N 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.
%F A292999 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).
%e A292999 For n=1: AAAA -> T(1,4)=1.
%e A292999 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.
%e A292999 Triangle starts:
%e A292999     1;
%e A292999     8,   10;
%e A292999    21,  120,     90;
%e A292999    40,  420,   1440,     840;
%e A292999    65, 1000,   6300,   16800,     8400;
%e A292999    96, 1950,  18000,   88200,   201600,    90720;
%e A292999   133, 3360,  40950,  294000,  1234800,  2540160,   1058400;
%e A292999   176, 5320,  80640,  764400,  4704000, 17781120,  33868800, 13305600;
%e A292999   225, 7920, 143640, 1693440, 13759200, 76204800, 266716800, ... .
%t A292999 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 *)
%Y A292999 Columns of the table: T(n,4) = A000567(n), T(n,5) = 10*A007586(n-1), T(n,6) = 90*A220212(n-2).
%Y A292999 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).
%K A292999 nonn,tabl
%O A292999 1,2
%A A292999 _Jeremy Dover_, Sep 27 2017