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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.

%I A292930 #10 Jul 23 2025 15:42:41
%S A292930 1,2,8,3,24,60,4,48,240,480,5,80,600,2400,4200,6,120,1200,7200,25200,
%T A292930 40320,7,168,2100,16800,88200,282240,423360,8,224,3360,33600,235200,
%U A292930 1128960,3386880,4838400,9,288,5040,60480,529200,3386880,15240960,43545600,59875200,10,360,7200,100800,1058400,8467200,50803200,217728000,598752000,798336000
%N 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.
%C A292930 Note that the three matching balls are necessarily the same color.
%F A292930 T(n, k) = binomial(k,3)*n!/(n+2-k)!.
%e A292930 n=1 => AAA -> T(1,3)=1;
%e A292930 n=2 => AAA,BBB -> T(2,3)=2;
%e A292930        AAAB,AABA,ABAA,BAAA,BBBA,BBAB,BABB,ABBB -> T(2,4)=8.
%e A292930 Triangle begins:
%e A292930   1;
%e A292930   2, 8;
%e A292930   3, 24, 60;
%e A292930   4, 48, 240, 480;
%e A292930   5, 80, 600, 2400, 4200;
%e A292930   ...
%o A292930 (PARI) T(n, k) = binomial(k,3)*n!/(n+2-k)!;
%o A292930 tabl(nn) = for (n=1, nn, for (k=3, n+2, print1(T(n,k), ", ")); print()); \\ _Michel Marcus_, Sep 29 2017
%Y A292930 Columns of table: T(n,3) = A000027(n), T(n,4) = A033996(n).
%Y A292930 Other sequences in table: T(n,n+2) = A005990(n+1).
%K A292930 nonn,tabl
%O A292930 1,2
%A A292930 _Jeremy Dover_, Sep 26 2017