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A123160
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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.
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2
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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, 1890, 1512, 462, 1, 49, 588, 2940, 7350, 9702, 6468, 1716, 1, 64, 1008, 6720, 23100, 44352, 48048, 27456, 6435, 1, 81, 1620, 13860, 62370, 162162, 252252, 231660
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| 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]
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REFERENCES
| Frederick T. Wall, Chemical Thermodynamics, W. H. Freeman, San Francisco, 1965 pages 296 and 305
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LINKS
| Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra 278, (2004), 342-359.
Laradji, A. and Umar, A. Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq. 7 (2004), 04.3.8
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FORMULA
| 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]
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EXAMPLE
| Triangle begins:
1
1, 1
1, 4, 3
1, 9, 18, 10
1, 16, 60, 80, 35
1, 25, 150, 350, 350, 126
...
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MAPLE
| 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
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MATHEMATICA
| 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]
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CROSSREFS
| Cf. A059481, A122899.
Sequence in context: A165732 A197698 A193011 * A039758 A109692 A157894
Adjacent sequences: A123157 A123158 A123159 * A123161 A123162 A123163
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KEYWORD
| nonn,tabl
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AUTHOR
| Roger Bagula (rlbagulatftn(AT)yahoo.com), Oct 02 2006
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Oct 26 2006 and Jul 03 2008
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