A289710 Triangle T(n,r) read by rows: order of the semigroup of orientation-preserving partial transformations of n elements with breath r.

1, 1, 1, 1, 4, 4, 1, 9, 27, 24, 1, 16, 96, 208, 128, 1, 25, 250, 950, 1325, 610, 1, 36, 540, 3120, 7290, 7416, 2742, 1, 49, 1029, 8330, 28665, 47922, 38563, 11970, 1, 64, 1792, 19264, 90720, 219968, 287168, 191808, 51424, 1, 81, 2916, 40068, 246078, 806274, 1509732, 1619676, 926073, 218718

Offset: 0

Author

R. J. Mathar, Sep 02 2017

Keywords

• tabl
• nonn
• easy

Examples

			1 ;
1 1;
1 4 4;
1 9 27 24;
1 16 96 208 128;
1 25 250 950 1325 610;
1 36 540 3120 7290 7416 2742;
1 49 1029 8330 28665 47922 38563 11970;
1 64 1792 19264 90720 219968 287168 191808 51424;
1 81 2916 40068 246078 806274 1509732 1619676 926073 218718;
		

Links

• A. Umar, Combinatorial Results for Semigroups of Orientation-Preserving Partial Transformations, J. Int. Seq. 14 (2011) # 11.7.5, Corollary 9, Table 2.1

Crossrefs

Cf. A092364 (column r=2), A289713 (row sums)

Programs

Maple

A289710 := proc(n,r)
    if r = 0 then
        1;
    else
        r*binomial(n,r)*binomial(n+r-1,n-1)-n*(r-1)*binomial(n,r) ;
    end if ;
end proc: