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A228617 T(n,k) is the number of s in {1,...,n}^n having shortest run with the same value of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A228617 #23 Jan 28 2019 18:16:17
%S A228617 1,0,1,0,2,2,0,24,0,3,0,240,12,0,4,0,3080,40,0,0,5,0,46410,210,30,0,0,
%T A228617 6,0,822612,840,84,0,0,0,7,0,16771832,5208,112,56,0,0,0,8,0,387395856,
%U A228617 23760,720,144,0,0,0,0,9,0,9999848700,148410,2610,180,90,0,0,0,0,10
%N A228617 T(n,k) is the number of s in {1,...,n}^n having shortest run with the same value of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%C A228617 Sum_{k=0..n} k*T(n,k) = A228618(n).
%C A228617 Sum_{k=0..n}   T(n,k) = A000312(n).
%C A228617 T(2*n,n)   = A002939(n) for n>0.
%C A228617 T(2*n+1,n) = A033586(n) for n>1.
%C A228617 T(2*n+2,n) = A085250(n+1) for n>2.
%C A228617 T(2*n+3,n) = A033586(n+1) for n>3.
%H A228617 Alois P. Heinz, <a href="/A228617/b228617.txt">Rows n = 0..140, flattened</a>
%e A228617 T(3,1) = 24: [1,1,2], [1,1,3], [1,2,1], [1,2,2], [1,2,3], [1,3,1], [1,3,2], [1,3,3], [2,1,1], [2,1,2], [2,1,3], [2,2,1], [2,2,3], [2,3,1], [2,3,2], [2,3,3], [3,1,1], [3,1,2], [3,1,3], [3,2,1], [3,2,2], [3,2,3], [3,3,1], [3,3,2].
%e A228617 T(3,3) =  3: [1,1,1], [2,2,2], [3,3,3].
%e A228617 Triangle T(n,k) begins:
%e A228617   1;
%e A228617   0,        1;
%e A228617   0,        2,    2;
%e A228617   0,       24,    0,   3;
%e A228617   0,      240,   12,   0,  4;
%e A228617   0,     3080,   40,   0,  0,  5;
%e A228617   0,    46410,  210,  30,  0,  0,  6;
%e A228617   0,   822612,  840,  84,  0,  0,  0,  7;
%e A228617   0, 16771832, 5208, 112, 56,  0,  0,  0,  8;
%Y A228617 Row sums give: A000312.
%Y A228617 Columns k=0-10 give: A000007, A228619, A228621, A228622, A228630, A228631, A228632, A228633, A228634, A228635, A228636.
%Y A228617 Main diagonal gives: A028310.
%Y A228617 Cf. A228154, A228273.
%K A228617 nonn,tabl
%O A228617 0,5
%A A228617 _Alois P. Heinz_, Aug 27 2013