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

0,5

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.

Links

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

Example

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;

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.

Sequence in context: A228273 A069521 A245687 * A119836 A158112 A219496

Adjacent sequences: A228614 A228615 A228616 * A228618 A228619 A228620

Keyword

nonn,tabl

Author

Alois P. Heinz, Aug 27 2013

Status

approved