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.

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, 6, 0, 822612, 840, 84, 0, 0, 0, 7, 0, 16771832, 5208, 112, 56, 0, 0, 0, 8, 0, 387395856, 23760, 720, 144, 0, 0, 0, 0, 9, 0, 9999848700, 148410, 2610, 180, 90, 0, 0, 0, 0, 10

Offset: 0

Author

Alois P. Heinz, Aug 27 2013

Keywords

• nonn
• tabl

Comments

Sum_{k=0..n} k*T(n,k) = A228618(n).

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

T(2*n,n) = A002939(n) for n>0.

T(2*n+1,n) = A033586(n) for n>1.

T(2*n+2,n) = A085250(n+1) for n>2.

T(2*n+3,n) = A033586(n+1) for n>3.

Examples

			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].
T(3,3) =  3: [1,1,1], [2,2,2], [3,3,3].
Triangle T(n,k) begins:
  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,  6;
  0,   822612,  840,  84,  0,  0,  0,  7;
  0, 16771832, 5208, 112, 56,  0,  0,  0,  8;
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

Row sums give: A000312.

Columns k=0-10 give: A000007, A228619, A228621, A228622, A228630, A228631, A228632, A228633, A228634, A228635, A228636.

Main diagonal gives: A028310.

Cf. A228154, A228273.