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 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. 5
 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; text; internal format)
 OFFSET 0,5 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)|). - Abdullahi Umar, Aug 25 2008 REFERENCES Frederick T. Wall, Chemical Thermodynamics, W. H. Freeman, San Francisco, 1965 pages 296 and 305 LINKS A. Laradji and A. Umar, Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra 278, (2004), 342-359. A. Laradji and A. Umar, Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq. 7 (2004), 04.3.8. FORMULA T(n,m) = If [n == m == 0, 1, n!*(n + m - 1)!/((n - m)!*(n - 1)!(m!)^2)]. T(n,k) = binomial(n,k)*binomial(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, Jan 24 2008 Or, T(n,k) = binomial(n,k)*(n+k-1)!/((n-1)!*k!. T(n,k)= binomial(n,k)*binomial(n+k-1,n-1). - Abdullahi Umar, Aug 25 2008 G.f.: (x+1)/(2*sqrt((1-x)^2-4*y)) + 1/2. - Vladimir Kruchinin, Jun 16 2015 From _Peter Bala, Jul 20 2015: (Start) O.g.f. (1 + x)/( 2*sqrt((1 - x)^2 - 4*x*y) ) + 1/2 = 1 + (1 + y)*x + (1 + 4*y + 3*y^2)*x^2 + .... For n >= 1, the n-th row polynomial R(n,y) = (1 + y)*r(n-1,y), where r(n,y) is the n-th row polynomial of A178301. exp( Sum_{n >= 1} R(n,y)*x^n/n ) = 1 + (1 + y)*x + (1 + 3*y + 2*y^2)*x^2 + ... is the o.g.f for A088617. (End) EXAMPLE Triangle begins:   1;   1,  1;   1,  4,   3;   1,  9,  18,  10;   1, 16,  60,  80,  35;   1, 25, 150, 350, 350, 126;   ... 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 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] max = 9; s = (x+1)/(2*Sqrt[(1-x)^2-4*y])+1/2 + O[x]^(max+2) + O[y]^(max+2); T[n_, k_] := SeriesCoefficient[s, {x, 0, n}, {y, 0, k}]; Table[T[n-k, k], {n, 0, max}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 18 2015, after Vladimir Kruchinin *) CROSSREFS Cf. A059481, A122899, A088617, A178301. Sequence in context: A197698 A193011 A214859 * A039758 A109692 A157894 Adjacent sequences:  A123157 A123158 A123159 * A123161 A123162 A123163 KEYWORD nonn,easy,tabl AUTHOR Roger L. Bagula, Oct 02 2006 EXTENSIONS Edited by N. J. A. Sloane, Oct 26 2006 and Jul 03 2008 STATUS approved

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Last modified February 29 08:26 EST 2020. Contains 332355 sequences. (Running on oeis4.)