A289714 Triangle T(n,k) read by rows: the number of semigroups of orientation-preserving partial transformations on n element with right waist k.

1, 1, 1, 1, 3, 5, 1, 7, 19, 34, 1, 15, 63, 135, 235, 1, 31, 191, 471, 911, 1556, 1, 63, 543, 1503, 3183, 5883, 9969, 1, 127, 1471, 4495, 10319, 20483, 37031, 62602, 1, 255, 3839, 12799, 31615, 67007, 128607, 229743, 388343, 1, 511, 9727, 35071, 92671, 208735, 423583, 796687, 1412863, 2389768, 1

Offset: 0

Author

R. J. Mathar, Sep 02 2017

Keywords

• nonn
• tabl
• easy

Examples

			1 ;
1 1 ;
1 3 5 ;
1 7 19 34 ;
1 15 63 135 235;
1 31 191 471 911 1556 ;
1 63 543 1503 3183 5883 9969 ;
1 127 1471 4495 10319 20483 37031 62602 ;
1 255 3839 12799 31615 67007 128607 229743 388343 ;
1 511 9727 35071 92671 208735 423583 796687 1412863 2389768 ;
1 1023 24063 93183 262143 625023 1336383 2638143 4894623 8637363 14621533 ;
		

Links

• A. Umar, Combinatorial Results for Semigroups of Orientation-Preserving Partial Transformations, J. Int. Seq. 14 (2011) # 11.7.5

Crossrefs

Cf. A289713 (row sums), A000225 (column 1), A053545 (column 2)

Programs

Maple

A289714 := proc(n,k)
    if k = 0 then
        1;
    else
        n*add(binomial(n-1,r-1)*binomial(r+k-2,r-1),r=1..n)-(n-2)*2^(n-1)-1 ;
    end if ;
end proc: