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%I #8 Sep 02 2017 15:52:13
%S 1,1,1,1,3,5,1,7,19,34,1,15,63,135,235,1,31,191,471,911,1556,1,63,543,
%T 1503,3183,5883,9969,1,127,1471,4495,10319,20483,37031,62602,1,255,
%U 3839,12799,31615,67007,128607,229743,388343,1,511,9727,35071,92671,208735,423583,796687,1412863,2389768,1
%N Triangle T(n,k) read by rows: the number of semigroups of orientation-preserving partial transformations on n element with right waist k.
%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
%e 1 ;
%e 1 1 ;
%e 1 3 5 ;
%e 1 7 19 34 ;
%e 1 15 63 135 235;
%e 1 31 191 471 911 1556 ;
%e 1 63 543 1503 3183 5883 9969 ;
%e 1 127 1471 4495 10319 20483 37031 62602 ;
%e 1 255 3839 12799 31615 67007 128607 229743 388343 ;
%e 1 511 9727 35071 92671 208735 423583 796687 1412863 2389768 ;
%e 1 1023 24063 93183 262143 625023 1336383 2638143 4894623 8637363 14621533 ;
%p A289714 := proc(n,k)
%p if k = 0 then
%p 1;
%p else
%p n*add(binomial(n-1,r-1)*binomial(r+k-2,r-1),r=1..n)-(n-2)*2^(n-1)-1 ;
%p end if ;
%p end proc:
%Y Cf. A289713 (row sums), A000225 (column 1), A053545 (column 2)
%K nonn,tabl,easy
%O 0,5
%A _R. J. Mathar_, Sep 02 2017
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