%I #8 Feb 20 2019 16:02:53
%S 1,1,1,1,3,2,1,7,8,6,1,15,24,28,22,1,31,64,96,112,90,1,63,160,288,416,
%T 484,394,1,127,384,800,1344,1896,2200,1806,1,255,896,2112,4000,6448,
%U 8952,10364,8558,1,511,2048,5376,11264,20160,31616,43392,50144,41586,1,1023,4608,13312,30464,59520,102592,157760,214656,247684,206098
%N T(n,k) is the number of order-decreasing and order-preserving partial transformations (of an n-chain) of waist k (waist(alpha) = max(Im(alpha))).
%H Laradji, A. and Umar, A., <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Umar/um.html">Combinatorial Results for Semigroups of Order-Decreasing Partial Transformations </a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.8. [From _Abdullahi Umar_, Oct 07 2008]
%F T(n,k) = (n-k+1)*Sum_{j=1..n} binomial(n,j)*binomial(k+j-2,j-1)/n for k > 0.
%F T(n,k) = 2*T(n-1,k) - T(n-1,k-1) + T(n,k-1) for n >= k >= 1; T(n,0)=1, T(n,1) = -1 + 2^n.
%e T(3,2) = 8 because there are exactly 8 order-decreasing and order-preserving partial transformations (of a 3-chain) of waist 2, namely: 2->2, 3->2, (1,2)->(1,2), (1,3)->(1,2), (2,3)->(1,2), (2,3)->(2,2), (1,2,3)->(1,1,2), (1,2,3)->(1,2,2).
%e Table begins
%e 1;
%e 1, 1;
%e 1, 3, 2;
%e 1, 7, 8, 6;
%e 1, 15, 24, 28, 22;
%e 1, 31, 64, 96, 112, 90;
%e 1, 63, 160, 288, 416, 484, 394;
%e 1, 127, 384, 800, 1344, 1896, 2200, 1806;
%p A145035 := proc(n,k) if k = 0 then 1; else (n-k+1)*sum(binomial(n,j)*binomial(k+j-2,j-1),j=1..n)/n ; end if; end proc: # _R. J. Mathar_, Jun 11 2011
%Y A006318 gives row sums of T(n, k).
%K nonn,easy,tabl
%O 0,5
%A _Abdullahi Umar_, Sep 30 2008