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%I #8 Sep 02 2017 15:55:04
%S 1,2,9,64,549,4566,34711,243944,1622025,10402858,65219931,403119036,
%T 2469129229,15032191422,91123985023,550590488656,3318118088721,
%U 19953003487314,119759533521571,717624299625188,4293868352394261,25658433286034662,153142720988745159,913049836619756664
%N The order of the semigroup of orientation-preserving or reserving partial transformations of n elements
%C Note the typo in corollary 27 of Umar (the 2nd equal sign in the equation in the first line should be a plus).
%H V. H. Fernandes, G. M. S. Gomes, M. M. Jesus, <a href="https://doi.org/10.1016/j.jalgebra.2008.11.005">Congruences on monoids of transformations preserving the orientation on a finite chain</a>, J. Alg. 321 (3) (2009) 743-757, proposition 1.10
%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 27, Tables 3.1-3.3.
%p # Table 3.3 of Umar
%p A := proc(n,k)
%p if k = 0 then
%p 1;
%p else
%p 2*n*add(binomial(k-1,p-1)*binomial(n-1,p-1)*2^(n-p),p=1..k)-(n-1)*2^n-1-(k-1)*2^(n-1)*binomial(n,2) ;
%p end if ;
%p end proc:
%p A289717 := proc(n)
%p add(A(n,k),k=0..n) ;
%p end proc:
%K nonn,easy
%O 0,2
%A _R. J. Mathar_, Sep 02 2017