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T(n,k) is the number of s in {1,...,n}^n having longest ending contiguous subsequence with the same value of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
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%I #30 Oct 07 2018 18:34:25

%S 1,0,1,0,2,2,0,18,6,3,0,192,48,12,4,0,2500,500,100,20,5,0,38880,6480,

%T 1080,180,30,6,0,705894,100842,14406,2058,294,42,7,0,14680064,1835008,

%U 229376,28672,3584,448,56,8,0,344373768,38263752,4251528,472392,52488,5832,648,72,9

%N T(n,k) is the number of s in {1,...,n}^n having longest ending contiguous subsequence with the same value of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%H Alois P. Heinz, <a href="/A228273/b228273.txt">Rows n = 0..140, flattened</a>

%F T(0,0) = 1, else T(n,k) = 0 for k<1 or k>n, else T(n,n) = n, else T(n,k) = (n-1)*n^(n-k).

%F Sum_{k=0..n} T(n,k) = A000312(n).

%F Sum_{k=0..n} k*T(n,k) = A031972(n).

%e T(0,0) = 1: [].

%e T(1,1) = 1: [1].

%e T(2,1) = 2: [1,2], [2,1].

%e T(2,2) = 2: [1,1], [2,2].

%e T(3,1) = 18: [1,1,2], [1,1,3], [1,2,1], [1,2,3], [1,3,1], [1,3,2], [2,1,2], [2,1,3], [2,2,1], [2,2,3], [2,3,1], [2,3,2], [3,1,2], [3,1,3], [3,2,1], [3,2,3], [3,3,1], [3,3,2].

%e T(3,2) = 6: [1,2,2], [1,3,3], [2,1,1], [2,3,3], [3,1,1], [3,2,2].

%e T(3,3) = 3: [1,1,1], [2,2,2], [3,3,3].

%e Triangle T(n,k) begins:

%e 1;

%e 0, 1;

%e 0, 2, 2;

%e 0, 18, 6, 3;

%e 0, 192, 48, 12, 4;

%e 0, 2500, 500, 100, 20, 5;

%e 0, 38880, 6480, 1080, 180, 30, 6;

%e 0, 705894, 100842, 14406, 2058, 294, 42, 7;

%e 0, 14680064, 1835008, 229376, 28672, 3584, 448, 56, 8;

%p T:= (n, k)-> `if`(n=0 and k=0, 1, `if`(k<1 or k>n, 0,

%p `if`(k=n, n, (n-1)*n^(n-k)))):

%p seq(seq(T(n,k), k=0..n), n=0..12);

%t f[0,0]=1;

%t f[n_,k_]:=Which[1<=k<=n-1,n^(n-k)*(n-1),k<1,0,k==n,n,k>n,0];

%t Table[Table[f[n,k],{k,0,n}],{n,0,10}]//Grid (* _Geoffrey Critzer_, May 19 2014 *)

%Y Row sums give: A000312.

%Y Columns k=0-4 give: A000007, A066274(n) = 2*A081131(n) for n>1, A053506(n) for n>2, A055865(n-1) = A085389(n-1) for n>3, A085390(n-1) for n>4.

%Y Main diagonal gives: A028310.

%Y Lower diagonals include (offsets may differ): A002378, A045991, A085537, A085538, A085539.

%Y Cf. A228154, A228617.

%K nonn,tabl,easy

%O 0,5

%A _Alois P. Heinz_, Aug 19 2013