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.

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, 115830, 24310

Offset: 0

Author

Roger L. Bagula, Oct 02 2006

Keywords

• nonn
• easy
• tabl

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

Examples

			Triangle begins:
  1;
  1,  1;
  1,  4,   3;
  1,  9,  18,  10;
  1, 16,  60,  80,  35;
  1, 25, 150, 350, 350, 126;
  ...
		

References

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

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened
• 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.

Crossrefs

Cf. A005773, A037965, A059481, A085373, A088218, A088617, A122899, A123164, A178301.

Programs

Magma

[Binomial(n,k)*Binomial(n+k-1,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 19 2022

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)];
Table[T[n, m], {n,0,10}, {m,0,n}]//Flatten
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 *)

SageMath

def A123160(n,k): return binomial(n, k)*binomial(n+k-1, k)
flatten([[A123160(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 19 2022

Formula

T(n, m) = n!*(n + m - 1)!/((n - m)!*(n - 1)!(m!)^2), with T(0, 0) = 1.

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

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)

From G. C. Greubel, Jun 19 2022: (Start)

T(n, n) = A088218(n).

T(n, n-1) = A037965(n).

T(n, n-2) = A085373(n-2).

Sum_{k=0..n} T(n, k) = A123164(n).

Sum_{k=0..floor(n/2)} T(n-k, k) = A005773(n). (End)

Extensions

Edited by N. J. A. Sloane, Oct 26 2006 and Jul 03 2008