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A123160 Triangle read by rows: T(n,k) = n!*(n+k-1)!/((n-k)!*(n-1)!*(k!)^2) for 0 <= k <= n, with T(0,0) = 1.

%I A123160 #44 Sep 16 2024 09:24:22
%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,
%T A123160 1120,1890,1512,462,1,49,588,2940,7350,9702,6468,1716,1,64,1008,6720,
%U A123160 23100,44352,48048,27456,6435,1,81,1620,13860,62370,162162,252252,231660,115830,24310
%N A123160 Triangle read by rows: T(n,k) = n!*(n+k-1)!/((n-k)!*(n-1)!*(k!)^2) for 0 <= k <= n, with T(0,0) = 1.
%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)|). - _Abdullahi Umar_, Aug 25 2008
%D A123160 Frederick T. Wall, Chemical Thermodynamics, W. H. Freeman, San Francisco, 1965 pages 296 and 305
%H A123160 G. C. Greubel, <a href="/A123160/b123160.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A123160 A. Laradji and A. Umar, <a href="http://dx.doi.org/10.1016/j.jalgebra.2003.10.023">Combinatorial results for semigroups of order-preserving partial transformations</a>, Journal of Algebra 278, (2004), 342-359.
%H A123160 A. Laradji and A. Umar, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Umar/um.html">Combinatorial results for semigroups of order-decreasing partial transformations</a>, J. Integer Seq. 7 (2004), 04.3.8.
%F A123160 T(n, m) = n!*(n + m - 1)!/((n - m)!*(n - 1)!(m!)^2), with T(0, 0) = 1.
%F A123160 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
%F A123160 T(n, k) = binomial(n,k)*(n+k-1)!/((n-1)!*k!).
%F A123160 T(n, k)= binomial(n,k)*binomial(n+k-1,n-1). - _Abdullahi Umar_, Aug 25 2008
%F A123160 G.f.: (x+1)/(2*sqrt((1-x)^2-4*y)) + 1/2. - _Vladimir Kruchinin_, Jun 16 2015
%F A123160 From _Peter Bala, Jul 20 2015: (Start)
%F A123160 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 + ....
%F A123160 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.
%F A123160 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)
%F A123160 From _G. C. Greubel_, Jun 19 2022: (Start)
%F A123160 T(n, n) = A088218(n).
%F A123160 T(n, n-1) = A037965(n).
%F A123160 T(n, n-2) = A085373(n-2).
%F A123160 Sum_{k=0..n} T(n, k) = A123164(n).
%F A123160 Sum_{k=0..floor(n/2)} T(n-k, k) = A005773(n). (End)
%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)];
%t A123160 Table[T[n, m], {n,0,10}, {m,0,n}]//Flatten
%t A123160 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_ *)
%o A123160 (Magma) [Binomial(n,k)*Binomial(n+k-1,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 19 2022
%o A123160 (SageMath)
%o A123160 def A123160(n,k): return binomial(n, k)*binomial(n+k-1, k)
%o A123160 flatten([[A123160(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 19 2022
%Y A123160 Cf. A005773, A037965, A059481, A085373, A088218, A088617, A122899, A123164, A178301.
%K A123160 nonn,easy,tabl
%O A123160 0,5
%A A123160 _Roger L. Bagula_, Oct 02 2006
%E A123160 Edited by _N. J. A. Sloane_, Oct 26 2006 and Jul 03 2008