A289710 - OEIS

%I #10 Sep 02 2017 15:46:05

%S 1,1,1,1,4,4,1,9,27,24,1,16,96,208,128,1,25,250,950,1325,610,1,36,540,

%T 3120,7290,7416,2742,1,49,1029,8330,28665,47922,38563,11970,1,64,1792,

%U 19264,90720,219968,287168,191808,51424,1,81,2916,40068,246078,806274,1509732,1619676,926073,218718

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

%H A. Umar, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Umar/umar2.html">Combinatorial Results for Semigroups of Orientation-Preserving Partial Transformations</a>, J. Int. Seq. 14 (2011) # 11.7.5, Corollary 9, Table 2.1

%e 1 ;

%e 1 1;

%e 1 4 4;

%e 1 9 27 24;

%e 1 16 96 208 128;

%e 1 25 250 950 1325 610;

%e 1 36 540 3120 7290 7416 2742;

%e 1 49 1029 8330 28665 47922 38563 11970;

%e 1 64 1792 19264 90720 219968 287168 191808 51424;

%e 1 81 2916 40068 246078 806274 1509732 1619676 926073 218718;

%p A289710 := proc(n,r)

%p     if r = 0 then

%p         1;

%p     else

%p         r*binomial(n,r)*binomial(n+r-1,n-1)-n*(r-1)*binomial(n,r) ;

%p     end if ;

%p end proc:

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

%K tabl,nonn,easy

%O 0,5

%A _R. J. Mathar_, Sep 02 2017