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

%I A289710 #10 Sep 02 2017 15:46:05
%S A289710 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 A289710 3120,7290,7416,2742,1,49,1029,8330,28665,47922,38563,11970,1,64,1792,
%U A289710 19264,90720,219968,287168,191808,51424,1,81,2916,40068,246078,806274,1509732,1619676,926073,218718
%N A289710 Triangle T(n,r) read by rows: order of the semigroup of orientation-preserving partial transformations of n elements with breath r.
%H A289710 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 A289710 1 ;
%e A289710 1 1;
%e A289710 1 4 4;
%e A289710 1 9 27 24;
%e A289710 1 16 96 208 128;
%e A289710 1 25 250 950 1325 610;
%e A289710 1 36 540 3120 7290 7416 2742;
%e A289710 1 49 1029 8330 28665 47922 38563 11970;
%e A289710 1 64 1792 19264 90720 219968 287168 191808 51424;
%e A289710 1 81 2916 40068 246078 806274 1509732 1619676 926073 218718;
%p A289710 A289710 := proc(n,r)
%p A289710     if r = 0 then
%p A289710         1;
%p A289710     else
%p A289710         r*binomial(n,r)*binomial(n+r-1,n-1)-n*(r-1)*binomial(n,r) ;
%p A289710     end if ;
%p A289710 end proc:
%Y A289710 Cf. A092364 (column r=2), A289713 (row sums)
%K A289710 tabl,nonn,easy
%O A289710 0,5
%A A289710 _R. J. Mathar_, Sep 02 2017