0,5

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]

Frederick T. Wall, Chemical Thermodynamics, W. H. Freeman, San Francisco, 1965 pages 296 and 305

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]

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

Or, T(n,k)=binom(n,k)*(n+k-1)!/((n-1)!*k!.

a(n,m) = If [n == m == 0, 1, n!*(n + m - 1)!/((n - m)!*(n - 1)!(m!)^2)]

T(n,k)= C(n,k)*C(n+k-1,n-1) [From A. Umar (aumarh(AT)squ.edu.om), Aug 25 2008]

Triangle begins:

1

1, 1

1, 4, 3

1, 9, 18, 10

1, 16, 60, 80, 35

1, 25, 150, 350, 350, 126

...

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[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]

Cf. A059481, A122899.

Sequence in context: A165914 A139621 A165732 this_sequence A039758 A109692 A157894

Adjacent sequences: A123157 A123158 A123159 this_sequence A123161 A123162 A123163

nonn,tabl

Roger Bagula (rlbagulatftn(AT)yahoo.com), Oct 02 2006

Edited by N. J. A. Sloane (njas(AT)research.att.com), Oct 26 2006 and Jul 03 2008