%I A123160
%S A123160 1,1,1,1,4,3,1,9,18,10,1,16,60,80,35,1,25,150,350,350,126,1,36,315,1120,
%T A123160 1890,1512,462,1,49,588,2940,7350,9702,6468,1716,1,64,1008,6720,23100,
%U A123160 44352,48048,27456,6435,1,81,1620,13860,62370,162162,252252,231660
%N A123160 Triangle read by rows: T(0,0)=1; T(n,k)=n!(n+k-1)!/[(n-k)!(n-1)!(k!)^2] 
               for 0 <= k <= n.
%C A123160 T(n,k) is also the number of order-preserving partial transformations 
               (of an n-element chain) of width k (width(alpha) = |Dom(alpha)|). 
               [From A. Umar (aumarh(AT)squ.edu.om), Aug 25 2008]
%D A123160 Frederick T. Wall, Chemical Thermodynamics, W. H. Freeman, San Francisco, 
               1965 pages 296 and 305
%D A123160 Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving 
               partial transformations. Journal of Algebra 278, (2004), 342-359. 
               [From A. Umar (aumarh(AT)squ.edu.om), Aug 25 2008]
%F A123160 T(n,k) = binom(n,k)*binom(n+k-1,k). The row polynomials (except the first) 
               are (1+x)*P(n,0,1,2x+1), where P(n,a,b,x) denotes the Jacobi polynomial. 
               The columns of this triangle give the diagonals of A122899. - Peter 
               Bala (pbala(AT)toucansurf.com), Jan 24 2008
%F A123160 Or, T(n,k)=binom(n,k)*(n+k-1)!/((n-1)!*k!.
%F A123160 a(n,m) = If [n == m == 0, 1, n!*(n + m - 1)!/((n - m)!*(n - 1)!(m!)^2)]
%F A123160 T(n,k)= C(n,k)*C(n+k-1,n-1) [From A. Umar (aumarh(AT)squ.edu.om), Aug 
               25 2008]
%e A123160 Triangle begins:
%e A123160 1
%e A123160 1, 1
%e A123160 1, 4, 3
%e A123160 1, 9, 18, 10
%e A123160 1, 16, 60, 80, 35
%e A123160 1, 25, 150, 350, 350, 126
%e A123160 ...
%p A123160 T:=proc(n,k) if k=0 and n=0 then 1 elif k<=n then n!*(n+k-1)!/(n-k)!/
               (n-1)!/(k!)^2 else 0 fi end: for n from 0 to 10 do seq(T(n,k),k=0..n) 
               od; # yields sequence in triangular form
%t A123160 t[n_, m_] = If [n == m == 0, 1, n!*(n + m - 1)!/((n - m)!*(n - 1)!(m!)^2)]; 
               a = Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[a]
%Y A123160 Cf. A059481, A122899.
%Y A123160 Sequence in context: A165914 A139621 A165732 this_sequence A039758 A109692 
               A157894
%Y A123160 Adjacent sequences: A123157 A123158 A123159 this_sequence A123161 A123162 
               A123163
%K A123160 nonn,tabl
%O A123160 0,5
%A A123160 Roger Bagula (rlbagulatftn(AT)yahoo.com), Oct 02 2006
%E A123160 Edited by N. J. A. Sloane (njas(AT)research.att.com), Oct 26 2006 and 
               Jul 03 2008